looking for clarification on a simple argument

So I'm sure anyone who likes probability and statistics enough has heard of this type of question. It goes something along the lines of:

A mother takes her son, out of two kids, to the park. What is the probability her other child is a girl?

And I would say 2/3. Because the 4 possible ways of having 2 kids are BB GG BG and GB. So the option GG is eliminated, and of the remaining 3 options another boy occurs once, a girl occurs twice.

Now on the flip side... if the question said:

A mother takes her** oldest **son, out of two kids, to the park. What is the probability her other child is a girl?

Again... the four possibilities are BB, BG GB and GG. Is birth order relevant to this question? Would the answer still be 2/3 or would it be 1/2 because the option BG is eliminated from the set (assuming girl is the older child in the option BG)