So we would write down the possibilities as (B-Tue, B-Tue), (B-Tue, G-Any), (B-Tue, B-NotTue) and the inverse for the latter two. This results in 27 cases. Of those 13 have two boys so the answer is 13/27.

Similarly if you had the knowledge “at least one is a boy born on May 11” then it would be very close to but slightly less than 50%.

So we can see in the limit as the information becomes more and more specific it turns into the unconditional probability. That is, the case of “the first is a boy, what is the probability both are boys” (50%).

I think this clarifies the situation in the OP pretty well.