My understanding is that the size $\alpha$ of hypothesis tests is defined as
$$ \alpha = \sup_{\theta \in \Theta_0} \mathbb{P}_\theta (X \in R) $$
where $\Theta_0$ is the subset of the parameter space associated with the null hypothesis $H_0: \theta \in \Theta_0$ and R is the rejection region such that
$$ \forall X \in R, \text{ reject } H_0 $$
So, in words, the size of hypothesis tests is the supremum of the probability to make type I error, i.e. rejecting $H_0$ when it is true. Correct?
Broadly yes, but $\Theta_0$ is not "the null hypothesis", but contains the set of parameters for which the null is true.
For example, when you test a composite hypothesis about the mean that $H_0:\mu\leq0$, then the null is true for any nonpositive $\mu$. The definition of the size now says that (a little heuristically in view of the proper definition of a $\sup$) when we compute the type-I error probabilities for all these nonpositive $\mu$, the largest of these probabilities, so the maximum type-I error probability, is $\alpha$.
