"The question writer took all sets of two child families and ruled out the bb case. Then they asked the exact question above" This is 1/3 chance - select gg from [gg,bg,gb]'

vs

"The question writer came across a girl from a two child family, then they asked the exact question above". This is 1/2 chance - select gg from [gg, gg, bg, gb] with gg listed twice since there's two ways to select a girl from that set; ie. coming across a girl is twice as likely to occur from the gg case than it is either gb or bg.

I think that's the clearest wording to get the message across. Either way it's the exact same question but it reasonably has a completely different answer. There's no way to resolve this ambiguity with the question as written.

That's a good framing. It's similar to the fact that the chance of a given star being in a multiple system (~47% in our vicinity [0]) is significantly higher than the chance of a given system having multiple stars (~30%), because counting by individual stars gives more weight to the multiple systems.

[0] https://astronomy.stackexchange.com/a/55505

Those questions are equivalent. What is important is the conditional “… given that I looked at a random child and it was a girl” / “… given that I looked at both children and at least one of them was a girl”.

The issue with asking whether the other is a girl is how you choose the first one.

If you look at one random child, see it's a boy and exclude the family, even though the other child may be a girl, then you get the 1/2 probability. If however in that case you also look at the second child, see that's a girl and consider the family anyway, then you get the 1/3