This statement leads me to believe you are still confused. Do you agree that if I know a family has two kids, I knock on the door and a boy answers and says "I was born on a Tuesday," that the probability of the second kid being a girl is 1/2? And in this case, Tuesday is irrelevant? (This the wikipedia called "sampling")

I agree with this.

Do you agree that if, instead, the parents give you the information "one of my two kids is a boy born on a Tuesday", that this is a different sort of information, information about the set of their children, and not about a specific child?

I agree with this if they said something along the lines of "One and only one of them was born on Tuesday". If not, I don't see how the Boy(tu)/Boy(tu) configuration has the same probability as the others, because it's twice as likely as the other two configurations that that is the configuration they are talking about when they say "One was born on Tuesday".

Here's my breakdown with 1000 families, to try to make it clear what I mean:

1000 Families with two children, 750 have boys.

Of the 750, 500 have one boy and one girl. Of these 500, 1/7, or roughly 71 have a boy born on Tuesday.

Of the 750, 250 have two boys. Of these 250, 2/7, or roughly 71 have a boy born on Tuesday.

71 = 71, so it's equally likely that there are two boys as there are a boy and a girl.

Having two boys doubles the probability that one boy was born on Tuesday compared to having just one boy.

And I don't think I'm confused about the sampling, because I didn't use the sampling reasoning to get my result*, but I'm not super confident about that so if I am just keep giving me numbers and hopefully it will click.

*I mean in the previous post, not specifically this post.