Short version of my response:
A mathematician poses a problem about a father with two children, mentioning that one child is known to be of a certain age and born on a certain day, and asks what the probability is that the other child is the same gender.
The answer to this question (assuming 50:50 birth rates) is 50%.
This is because I'm accumulating probability over sets of questions & fathers & children without omitting any cases, much as the non-50% answer accumulates probability over sets of children and omits some cases.
Long version:
If you try to understand probability without trying to count over large sets of examples (i.e. statistics), you will almost surely make mistakes.
Let's step back to just the Two Children Problem and, I'm going to try to point out what I hope is a slightly deeper intuition about the two answers.
The naive answer of 1/2 just assumes the child gender is independent. Woohoo, rock on.
The "clever" answer imagines that we have thousands of fathers with two children. Some of those fathers are the fathers of two girls. The rest are the fathers of at least one boy. Only the latter fathers can pose this problem; therefore, if we randomly select a father from that set of fathers who can pose this problem, we have only a 1/3rd chance of drawing a father who is the father of two boys.
But we're linking the fathers to the problem in an overly specific way. Suppose we draw a father randomly from the set of all fathers with two children. That father may have two boys, two girls, or one of each. If we draw a father with at least one boy, we can proffer him to the reader and ask what the odds are his other child is a boy. If we draw a father with at least one girl, we can proffer him to the reader and ask what the odds are his other child is a girl.
So, if you actually had to do this--pick one person, ask the question, and you're done--what would you do? Well, it's dumb to draw a random father and then go "oh crap, we can't actually ask a question, this whole thing was a waste of time". So you'd probably ask the "at least one boy" question if they had two boys, the "at least one girl" question if they had two girls, or... one or the other question in the other case. Maybe you always ask the boys question. Maybe you randomly pick. If your tie-breaker is to just always ask "at least one boy" if it's possible, then the answer to "at least one boy" is 1/3 chance the other is a boy, but the answer to "at least one girl" is a 100% chance the other is a girl!
If you choose "fairly", then when you draw a father with both a boy and a girl, then half the time you ask "at least one boy" and half the time you ask "at least one girl".
If you choose that way, and happen to draw a father with the "at least one boy" question, then the probability the other child of his is a boy is 50%. (That is, given 100 fathers, the three cases are 2 girls: 25; 2 boys 25; 1 each 50; you split the 1 each case between the 2 questions, so you have 50 cases where you ask "at least one girl"--25 of those are they have 2 girls, 25 of those they have one of each; and 50 cases where you ask "at least one boy" with the same odds.)
Thus a complete analysis of all the cases under a more plausible situation for where you might be asking this question returns to the naive solution: the other child will be a boy 50% of the time.
So, as other people have said it depends on context. If the context is that it's a mathematician posing the question once about a hypothetical non-real situation, and the mathematician is sexist or otherwise favors mentioning boys, then maybe it's 1/3rd. But if he's not--if he chose the question randomly based on the gender choices available to him--then it's 50%.
Moreover, even if you choose a sexist bias for how you split the middle cases, if you measure over asking ALL questions, both "at least one boy" and "at least one girl", no matter how you split up the cases, the average odds that the other child is the same gender as the listed gender is 50%.
The same analysis applies to the tuesday problem; if you pick a random two-child-parent and ask some similar question--rather than simply failing to ask a question if the desired question doesn't apply--the average over all such answers will be 50%.