The way to see this is bayes rule. p(answer | data) = p(data | answer) * p(answer) / (sum_{all possible answers'} p(data | answer') * p(answer')). So for this question, that's expands to:

    p(both are girls | you're told at least one is a girl)
     = p(you're told at least one is a girl | both are girls) * p(both are girls) / (
            p(you're told at least one is a girl | both are girls) * p(both are girls)
            +
            p(you're told at least one is a girl | they aren't both girls) * p(they aren't both girls)
        )

This is the crux of the thing. Different readings of the setup imply different answers to p(what you're told | the unknowns).

It's also a great case of where bayes rule shorthands can be slippery. You'll usually abbreviate it out (hell, it was tedious to write this way even with copy-paste). But if you abbreviate "you're told there's at least one girl" to "there's at least one girl", then you've stopped modeling a crucial part of the setup. p(there's at least one girl | they aren't both girls) has an unambiguous answer.