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gjm comments on Probability puzzles - Less Wrong
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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.
Indeed, and this exact malformed problem is also discussed in the post "My Bayesian Enlightenment":
A more general lesson is that whenever the answer to a puzzle causes you to go, "oh how wondrous that this question could have such a strange answer", you were probably tricked into accepting an anti-helpful framing of the problem, and one of the reasons why the puzzle-poser didn't guide you into a helpful framing instead was probably exactly that such anti-helpful framings cause people to feel that way.
This post is not evidence for that lesson. When OP's puzzle is stated as intended it indeed has a wonderful and strange answer. The meta-puzzle: "Are these two puzzles essentially the same?" referring to the puzzle as intended and as presented also has a wonderful and strange answer; in fact, John Baez and maybe all of his commenters have been getting it wrong for several years. Our intuition is imperfect, and whether the puzzles you come across tend to use this fact or just trick you with sneaky framing depends on where you get your puzzles.
Well, another general lesson. I think it's largely orthogonal to the general principle I mentioned. (For the avoidance of doubt, I agree with yours too.)