If you make the assumptions I made, the maths is unambiguous. Namely, I assumed that a child has a 50/50 probability of being born a boy or a girl. I assumed the child who opens the door is random. I don't think I made any other assumptions that could have been made any other way. With those assumptions, I'm pretty confident my answer is the only correct one, though I'd love to see an argument otherwise.

If the probability of a child being a girl is different, say, 52%, that will affect the result.

More interestingly, if the probability of which child opens the door is different, that will affect the result. If there's a 100% chance the elder child opens the door, it goes to 50/50 of the gender of the second child. This makes it like the "coin on the left" example you gave.

If we said the elder child is going to open the door 75% of the time…well, the maths becomes more complicated than I can be bothered with right now. But it's an interesting scenario!