"A family has two children, at least one of whom is a boy; what is the probability that both are boys?"
and
"A family has two children, at least one of whom is a boy named Bartholomew; what is the probability that both are boys?"
have different answers.
It depends a great deal on how you extracted the information in the latter case. If you found this out in answer to the question, "Do you have a boy named Bart?" you are quite correct. If, however, you found this out in answer to the question, "Tell me the name of one of just one of your boy(s)," (to which the answer is Bart) then I assert you've been given no additional information.
In your posted analysis, those families with two boys, one of whom is named Bart, only half of them would answer Bart to the second question, while all would answer yes to the first. The problem here is what assumptions you are likely to make having heard the underspecified problem. When you hear, "A family has two children, at least one of whom is a boy," one tends to think that you are getting the answer to, "Do you have at least one boy?" and not "What is the sex of one of your children?" Because Bart is such an unusual name, it's more natural to think that you are receiving the answer to the question, "What is the name of one of your boy(s)?" In short, these questions are notoriously difficult not so much because the probability is hard, but because it's hard to pin down the hidden assumptions. David