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It matters if the assumption is that possibility of each child being boy/girl is 50%.

The first child can be boy or girl. The second child can also be boy or girl. So the probabilities are like this: BB: 25% GG: 25% BG: 50%

You can replace older/younger on the article with child 1/child 2.

That GB is twice as likely as either GG or BB. For this reason, reference is made to BG to make this more apparent.

You only misunderstand because the author is presenting it stupidly (as you state, incorporating order where order is irrelevant), leading to his own stupid conclusion. Read my post elsewhere in the thread for the full explanation, and if you find this kind of thing interesting, look up the Monty Hall problem (the point of which is that changing the information changes the odds).