The graph you've sketched looks a little like this graph of incidence rates* :

Assuming that this is the graph you're thinking of, it shows the number of people per year per 100,000 persons diagnosed with diabetes. (In this case for the period 2001-2015, and pooled into age categories 5 years wide.)

I figure that this is the graph of the PDF whose associated random variable returns the age at which you will develop the disease, assuming you will develop the disease in the first place;

Not quite, though it is closely related. It is the graph showing the proportion of people of any given age and gender who will be first diagnosed with diabetes in the next year. For example, approximately 25 in every 100,000 fifty-five-year-old men will be diagnosed with diabetes.

So assuming that diagnostic criteria, treatment options, and population dynamics don't change much in the coming decades (a very dubious assumption, but I can't predict the future so I have to work with past data), then: Conditional on you reaching the age of 55, your odds of being diagnosed with diabetes before the age of 56 are 25/100000.

This is a little higher than the probability that will be diagnosed for the first time at age 55. You could die before reaching 55. You could be diagnosed at an earlier age, and so be excluded from that category. To convert this chart into the PDF you want, you would need to account for both of these effects.

* (Rogers, Mary & Kim, Catherine & Banerjee, Tanima & Lee, Joyce. (2017). Fluctuations in the incidence of type 1 diabetes in the United States from 2001 to 2015: A longitudinal study. BMC Medicine. 15. 10.1186/s12916-017-0958-6.) (Licensed under CC-BY. My thanks to the authors for releasing it under open access.)

The graph you've sketched looks a little like this graph of incidence rates* :

Assuming that this is the graph you're thinking of, it shows the number of people per year per 100,000 persons diagnosed with diabetes. (In this case for the period 2001-2015, and pooled into age categories 5 years wide.)

I figure that this is the graph of the PDF whose associated random variable returns the age at which you will develop the disease, assuming you will develop the disease in the first place;

Not quite, though it is closely related. It is the graph showing the proportion of people of any given age and gender who will be first diagnosed with diabetes in the next year. For example, approximately 25 in every 100,000 fifty-five-year-old men will be diagnosed with diabetes.

So assuming that diagnostic criteria, treatment options, and population dynamics don't change much in the coming decades (a very dubious assumption, but I can't predict the future so I have to work with past data), then: Conditional on you reaching the age of 55 and not already being diagnosed, your odds of being diagnosed with diabetes before the age of 56 are 25/100000.

This is a little higher than the probability that will be diagnosed for the first time at age 55. You could die before reaching 55. You could be diagnosed at an earlier age, and so be excluded from that category. To convert this chart into the PDF you want, you would need to account for both of these effects.

* (Rogers, Mary & Kim, Catherine & Banerjee, Tanima & Lee, Joyce. (2017). Fluctuations in the incidence of type 1 diabetes in the United States from 2001 to 2015: A longitudinal study. BMC Medicine. 15. 10.1186/s12916-017-0958-6.) (Licensed under CC-BY. My thanks to the authors for releasing it under open access.)