When is it important for a practitioner to understand CIs?

I think the interpretation you provide is harmless in most basic research scenarios -- wrong, but ultimately harmless $^1$. Indeed, I have been known to provide this interpretation for non-technical stakeholders just to avoid having to walk them through the idea of repeated sampling.

However, if you do use this interpretation you need to be sure to combat misinterpretations of the CI as a prediction interval. For example, were you to talk about a confidence interval for the mean in this way, and since the mean is used as a prediction, the less statistically inclined may think that "There is 1-a probability the next observation is in my CI" which would be both wrong and harmful.

More to your question, so long as the physician is just doing basic research -- running regressions, making table 2, publishing the paper -- then this interpretation is fine. As soon as decisions and policy are being changed, then I would probably insist on some more nuanced understanding of the machinery being used to make those decisions since there is more at risk.

$^{1.}$ Actually not even that wrong, depending on what school of thought you subscribe to. I am told Confidence, Likelihood, Probability: Statistical Inference with Confidence Distributions explains this perspective more thoroughly, but I admit I have not read it myself.

Greenland has proposed to rename confidence intervals to "compatibility intervals", which he defines as "the values of the parameter which, when combined with the background assumptions, produce a test model that is 'highly compatible' with the data". This is a nice way to report the values computed as the confidence interval, without using the often-confusing term "confidence".

Let's say we test a new drug compared with a placebo in a randomized controlled trial to see whether the drug lowers blood pressure. After 12 weeks on the drug, the systolic blood pressure is on average 30 mmHg lower (95% CI 15 to 45 mmHg). Let's look at 4 different scenarios:

1. The drug we tested was a homeopathic preparation that was so diluted that not a single molecule of the original substance should remain (or we alternatively realize that we accidentally gave placebo to everyone).
2. This was the first small trial for this drug in patients, the mode of action is completely novel and we know that new drugs in Phase 1b/2a have a success rate of about 10% (with many of the 90% that don't make it failing due to a lack of efficacy).
3. This was a small Phase 4 study of a drug that was shown to lower blood pressure by 10 mmHg (95% CI 7.5 to 12.5 mmHg) based on a meta-analysis of two very large Phase 3 studies in the same population and a similar study design. While the Phase 3 studies included countries such as Belgium, Portugal, Norway, Finland and Germany, this new study involves the Netherlands, Spain, Sweden, Denkmark and Austria.
4. This was the second Phase 3 study for a drug and the first Phase 3 study already had a point estimate of 31 mmHg blood pressure lowering (95% CI 14 to 47 mmHg). The drug is also from a class of drugs, for which 3 others drugs are already approved and showed similar effect sizes in their Phase 3 studies.

If you believe there's a 97.5% probability the true blood pressure lowering effect of the drug compared with placebo in the study population is >15 mmHg in all the cases (and similarly exactly a 2.5% probability that it's > 45 mmHg, but in a sense that's less interesting), you run the risk of making a lot of bad medical and business decisions.

Most reasonable people would argue that in the first scenario, you should probably have near zero probability that the true treatment effect of the intervention is >15 mmHg, in the second & third scenarios it's higher but substantially below 97.5% (perhaps lower in the third scenario vs. the second, but one could disagree), and substantially above 97.5% in the fourth scenario.

You may argue that I'm going all Bayesian on you, but it's extremely rare that we don't have some prior knowledge. Arguably, a Bayesian perspective might allow you to claim the kind of "95% probability the true value is in the CI" statement, but that's only if an (almost) completely flat prior is correct (an effect of blood pressure lowering of near 299,792,458 is just as likely as one near 3.141592654 or $- 17.213 \times 10^{-231321}$). When we don't have absolutely no prior information, this is incorrect (and shrugging your shoulders and refusing to pin down the available prior information because it's too tedious to work out, is not an excuse, either).

There are already good answers, so the following is more of a commentary or counterquestion: but if you find the quasi-Bayesian interpretation of confidence intervals so much more intuitive, why not simply use a Bayesian approach all the way?

If you want an interval that can be interpretated as "the probability that $\theta$ is within this range is $(1-\alpha)\%$", why not simply calculate the statistical object that allows you to make such statements unambiguously?

What is the precise definition of a confidence interval?

I made a video about this aimed at non-statisticians: https://youtu.be/jrUrjv_yM0M

A brief summary:

1. A procedure for calculating a 95% confidence intervals will result in [an interval containing the true parameter] 95% of the time.
2. In theory, you could construct confidence intervals in multiple ways to satisfy (1), but all result in (slightly) different intervals.
3. It is possible for one confidence interval to be shorter and completely contained in another interval, even though both were made with a valid 95% confidence procedure.
4. Hence, the statement of probability is about the procedure used to create intervals, not about any particular interval itself.

Below is the correct interpretation visualized. Imagine performing 100 experiments and constructing 95% confidence intervals each time:

(On average) 95% of these intervals will contain the true value.

When does the precise definition of a confidence interval matter?

In my opinion, this is especially the case...

• When it can be logically expected to differ substantially from a credible interval (strong prior beliefs, small data).
• When it is used in subsequent calculations. In this case, it is important to know that the bounds of a given interval do not necessarily provide protection up to a specified probability.
• When writing a scientific article. Misinterpretations propagate when authors are imprecise with language.

It's all about the expression. To put it very simple (because there is no more to it)

(shortcut: familiar with p-values? It's just the opposite)

• A confidence interval is a "guess"

• The guess is correct with probability 1 - $\alpha$

The crucial point is that the probability is associated with the "guess" that you make: whatever procedure you choose to make your guess, if repeated many times, the guess coming from the procedure will be correct a certain fraction of times.$^1$

That means, if you actually make a guess at something, you know, from trying it out often before, your guess will be right with a certain probability.

why does it matter?

Inference using data is about finding out, how was this data produced; it's about infering an underlying quantity of the data production mechanism (such as a natural constant, or the fact that medicin works or not). Your data is one out of infinitely many possible samples. But the value that we try to infer is already determined.

$^1$ That's the probability that "the procedure yields a guess that is correct", in short, "that your guess is correct"; in correct: the coverage of your interval.

I am going to give a non-medical take.

As an engineer, I can look at historic hourly temperatures at a site, and determine what a change in hardware would have given them if it had been in place in that period.

"If you had the economizer in last year you would have saved $\$$xyz on your utilities at a price of $\$$pqr"

Last year in weather is not a perfect indicator for this year in weather, so saying "you should expect to save $\$$xyz this year if you implement it" is a nearly always wrong. When you estimate the mean, then half the time the guess is too low, and the other half it is too high.

So a better approach is to bootstrap resample the temperatures for a few years, and look at the quantiles of the savings, and give a range to go with the central tendency. I like to say something to the effect of "your range should be between $\$$abc and $\$$def, and though the best single value to $\$$xyz, about half the time the center is going to be too low and the other half of the time too high, but in general your values should be within the band as long as the last few years are consistent indicators of the next few years when it comes to weather."

That shows how it can be useful in practice, and turn something that is wrongly understood by the customer 100% of the time into something where you are not wrong about 95% of the time, given your caveat of historic consistency.

For example, he finds the following confusing: Denote the parameter of interest θ . Before data are drawn, P(θ∈CI)=1−α.

That is indeed confusing, to the point of communicating something false. Wording it that way makes it sound like we have a fixed CI, and $\theta$ is a random variable that can be in the CI or not. But in fact $\theta$ is fixed, and the CI is a random variable that can include $\theta$ or not. While "The CI includes $\theta$" and "$\theta$ is in the CI" are logically equivalent, they communicate different things, especially when we start talking about the probability of the events. When we say that we have a 95% CI, what we mean is that we have an interval that was generated by an algorithm that (given certain assumptions about the distribution) has a 95% probability of producing an interval that contains the true parameter.

Suppose you know someone who, when they tosses a horseshoe, gets it on the stake 95% of the time. If you can see the horseshoe after they throw it, but not the stake, then the horseshoe is a 95% CI for where the stake is.

However, once data are drawn there isn't any probability anymore and the true parameter is either inside the CI or not.

No. Not "The true parameter is either inside the CI or not", but rather "The CI either contains the true parameter or not."

An example where misunderstanding CIs may lead to wrong clinical decisions would be ideal.

Two issues (other than having a flawed model) are confusing P(A|B) and P(B|A), and failing to incorporate additional data.

What sort of situation would a physician rely on CIs? One I can think of is if there's a diagnostic test that estimates some quantity, and there's some cutoff for diagnosing a disease, and the test yields a CI that excludes that cutoff. One might then conclude that there's only a 5% change that the "real" value is past the cutoff. But remember that a 95% CI means that 95% of all of the CIs include the true value. CIs that exclude the cutoff are a subset of all CIs. So it doesn't necessarily follow that 95% of those CIs exclude the true value.

Going back to the example of someone throwing horseshoes that I gave earlier, if I tell you that they get the horseshoe on the stake 95% of the time, then if you have nothing other than this to go on, you can be confident that there is a good chance that it's on the stake now. But what if I then tell you that actually, they do 95% of their horseshoe throwing during the day, and get 100% of those throws onto the stake, but get 0% of their nighttime throws onto the stake? Now it's critical to know what time of day it is.

Similarly, if a patient has a bunch of symptoms that are pointing towards a disease, but one test gives a 95% CI that excludes the cutoff, those other symptoms shouldn't be ignored in favor of the one test. And if all you have is the test, you should think about whether you could gather further information, rather than just stopping at "95% chance of healthy!"

For example, he finds the following confusing: Denote the parameter of interest $\theta$. Before data are drawn, $$ \mathbb{P}(\theta \in CI) = 1-\alpha. $$

However, once data are drawn the is no probability anymore and the true parameter is either inside the CI or not.

This is a misintepretation of the confidence interval, one which confuses things further.

The x% confidence interval is precisely defined with respect to observed data; it is the range of 'models' (or parameterisations, if you prefer) for an appropriately stated hypothesis type, which would fail to reject the null hypothesis at the x% significance level "given" the observed data.

So it doesn't make sense to talk about a confidence interval before observing data. A CI is not an expression of posterior probability for the presence of a parameter over an interval: the name for this statistic is "credible interval", which is a very different concept.

The statistical guarantee a CI makes instead, is that, if you were to repeat an experiment an infinite number of times, such that you obtain new data from the same population from which you calculated a new x% CI each time, then x% of the time your computed CIs would correctly contain the true population parameter.

One way to think about this is as follows: As someone else mentioned here, effectively, in the same way that the p-value is effectively a measure of compatibility between a given model and a given set of data, then the CI effectively gives you the range of models that are sufficiently compatible with the observed data. When you repeat an experiment and get slightly different data, you'll get a slightly different CI, because the range of compatible models will be slightly different. Every now and then, with low probability, your data might be an outlier, and atypical of the actual population, and obtained simply by chance. In that case, your CI may show a range of models that wouldn't actually be very compatible with the population. But this CI will be obtained rarely; as rarely as it is rare to get an atypical sample from the population by chance.

Given the difficulty that experienced statistical practitioners have when interpreting confidence intervals (see answers and comments here for evidence), it seems to me unlikely that there would be circumstances where such misinterpretation by a medical practitioner would be important. Therefore my answer to the titular question is "never".

That then carries with it some doubts about whether confidence intervals should be used (and taught) as the go-to type of interval statement.

Your question touches on the classic (should I say "purist"?) frequentist interpretation of CI's. @Demetri Panaos says that your interpretation is wrong (but maybe not so much, as his footnote states), but harmless. I will go one step further. Your interpretation is correct, and exactly equivalent to the "accepted" interpretation. So let me share how I explain this to my (undergrad) students (who seem to get it).

Let's start with the "official" interpretation. If I were to take a very large number of samples, and for each compute its associated CI (of whatever parameter $\theta$ I am interested in), I would get an equally very large number of CI's, all somehow different (because of the randomness of my variable). These CI's are such that $100\cdot(1-\alpha)%$ of them would actually contain the true population parameter $\theta$.

Now, let's continue from here. The fact is you do not have a very large number of CI's, but only the 1 from your experiment. But that 1 has been taken randomly out the very large number of (theoritical) CI's one could have obtained. So that one, single CI has a $100\cdot(1-\alpha)%$ chance of containing the true population parameter $\theta$ (because you randomly took it from a population which had a $100\cdot(1-\alpha)%$ probability of containing $\theta$).

Now your friend asks; "But after I got my sample, my single CI either contains $\theta$ or it does not". This is true. But we do not know $\theta$; there is uncertainty about its true value. Hence we can use probabilities to describe this uncertainty: there is a $100\cdot(1-\alpha)%$ probability that this CI contains the true $\theta$ (the CI is a random variable, from a population with $100\cdot(1-\alpha)%$ proportion of success).

When the purists get very upset is if you start saying "$\theta$ has a $100\cdot(1-\alpha)%$ of lying in the CI" (because $\theta$ is not a random variable, and hence talking about probabilities associated with $\theta$ is somehow incorrect; but the above sentence does not say that $\theta$ is random, only that its belonging to this 1 CI is random).

So the "correct" interpretation is "there is a $100\cdot(1-\alpha)%$ probability that this CI contains the true $\theta$", and an incorrect one is "$\theta$ has a $100\cdot(1-\alpha)%$ of lying in the CI". To which I say "distinction without a difference". The first sentence says "there is a probability that B contains A", the second says "there is a probability that A is contained in B". These 2 sentences are semantically exactly equivalent. One is in the active voice, the 2nd in the passive voice. While English teachers have taught us not to use the passive voice, this is math/statistics here, and both statements are logically 100% equivalent.

So not only is your interpretation harmless, it is just as valid. $P(\theta ∈ CI)=1−\alpha \Longleftrightarrow P(CI ∋ \theta)=1-\alpha$
Both express the exact same uncertainty about the relative appertenance of $\theta$ to CI.

And therefore, as you ask at the end of your question, "an example where this type of misunderstanding of CIs may lead to wrong clinical decisions" does not exist.
(there could be several other misunderstandings which could lead to wrong clinical decisions; e.g. confusing the $\alpha$ error rate -which is controlled by the CI, for the PPV, aka false discovery rate. But that is for another answer...)