I would translate this as "The girl chooses another cake" (not accepted, reported). Saying "a different cake" is ambiguous, it could mean just another cake (what the Hebrew sentence means), but it could also mean a cake that's somehow different (e.g. of a different type), which could be implied in this Hebrew sentence, but you'd probably say עוגה שונה or עוגה מסוג אחר or something.
Yan Nasonov (Ynhockey), I disagree. There's more ambiguity in the answer that you prefer than in the recommended answer.
Saying "a different cake" implies a desire for an alternative, rather than an addition, to one that had been considered previously. Although it makes no implication about the nature of the difference, doing so is not its purpose, because this isn't the entire story, so that is merely an omission, not an ambiguity.
Saying "another cake" IS ambiguous as to whether the girl rejects the first cake or chooses an additional one; and in either case, it has the same omission of information about the new choice.