> ... in the table, he DOES tell you that in the BG case.
  > Otherwise there aren't two spots; there's only one.  He
  > introduces the older/younger thing, then fails to apply
  > the new information across all cases.

I'm having real trouble understanding you here. The explanation lists all the possibilities, and to do so it's necessary to distinguish between the children. The most obvious way to do that is to talk about the older and younger.

we know that in families with two children about half the time you have one of each sex. You only get that if you distinguish between the children in some sense so that there are four overall possibilities: BB, BG, GB, GG. If you don't distinguish between the children then there are only three possibilities: Both boys, both girls, one of each. Doing real world trials clearly shows that model to be flawed. We must distinguish between the children when enumerating cases.

Forgive me if this is all obvious to you, but I honestly can't see your argument, so it's necessary to lay down much more detail to try to find where your reasoning varies from mine.

So now consider the situation I laid out. Take all families with two children. There are four equally likely possibilities. Eliminate those who cannot truthfully say "At least one child is a boy." You are left with three equally likely possibilities. In only one of those do we have two boys. Thus one out of three possibilities has both children boys.

The probability of both children being boys is 1/3.

Can you explain where that reasoning is faulty?

Also, computer simulations clearly show that under this model of what's going on, the chance of two boys is 1/3. If you explain the faulty reasoning, you'll also need to explain why the computer simulation gives the same answer.