The coins example is no different to the boy/girl problem.
Where you are confused Chrome Prince is that the order IS important.
The sample space is in fact -

Boy/Boy
Boy/Girl
Girl/Boy
Girl/Girl

Where each outcome is a 1/4 or 25% chance of occuring.

So if the order is not spectific there is a 1/4(25%) chance of having 2 Boys, a 1/4(25%) chance of having 2 Girls and a 2/4(50%) chance of having a Boy/Girl (any order).


Now the problem is conditioned on the fact that there is "at least one girl" so the Boy/Boy scenario can be eliminated fom the sample space leaving -

Boy/Girl
Girl/Boy
Girl/Girl

Where each outcome is a 1/3 or 33% chance of occuring. Given that we have already singles out a Girl, we are left with a Boy in 2 out if the 3 possibilities. This leads to the 2/3 Boy and 1/3 Girl which happens to be the correct answer.

The hard part to comprehend is the wording of the question.