The correct answer to puzzle 1, as posed, is not in fact simply 1/3, because you've got to factor in Pr(you say "I have two children and one is a boy" | each scenario) and it's not at all clear that these are equal in the BB, BG, GB cases. For instance, if you say that in the ordinary course of events (rather than, e.g., to pose a puzzle) I think BG and GB are very much more likely than BB, because if you had two boys why on earth would you say "I have two children and one is a boy"?

There's a general principle here: when you discover something new (by, e.g., being told something, or seeing something), the correct information to update on is not what you've been told, or seen but the fact that you've been told, or seen, it. In some cases this doesn't matter. In many others (including, but not limited to, those where you have reason to doubt the accuracy of what you've been told or seen) it does.