His point is that a person telling you they have one girl would be accepted as referring to a particular child. Thus, BG and GB merge and the odds are 50% of a boy and a girl.
Pedantically, they don't merge, you just rule out GB or BG. So, initial conditions:
1) X: G, Y: G
2) X: G, Y: B
3) X: B, Y: G
4) X: B, Y: B
Which still seems a little weird to me, that knowing which is a girl, regardless of which one you know about, changes the chances.
Exactly - the difference is between "at least one is a girl" and pointing out a specific child as a girl.
The only real objection to the problem as it's posed is that it's pretty unlikely that anyone would say that one of their children was a girl when both of their children are girls. For a normal conversation, the ambiguity wouldn't be present; "two children" combined with "one is a girl" means the chances of one boy and one girl are 100%.
It's a bit like the old joke about "which month of the year has 28 days?" - answer being, they all have 28 days, just some have more.
However, in other less contrived situations, this is important. People risk discounting permutations, and only considering combinations, when the permutations are necessary to get a correct view of the odds.