Why can't you just disregard the existing boy and reframe the question as the probability that the other child is a boy, and the space of all possible answers is BG and BB, equally probable (1/2)? Not really following explanations I find online.
No matter how I frame or interpret this question, the birthdays and birth order appear completely irrelevant - the results are still ~0.50 as expected. Whatever the author was trying to say, they didn't communicate it well. I'm really curious exactly what word or phrase the author thought I was supposed to take to mean something else. If someone could edit one of these simulations to show what the author intended then that would probably be the clearest way.
