Which answer is "correct" depends on your interpretation of the question, as you say, and that's exactly the point. Like many paradoxes of the kind, it illustrates the ambiguity of the question and, by extension, the lack of nuance in the way we describe and reason about probabilities (in general, that is; of course one who has learned some probability theory is likely to be able to understand and communicate the difference).
Implicit in any answer is a way of valuing the choices made. In the 1/3 case, it is assumed that being right twice is better than being right once - for example, you get $100 each time you answer correctly. In the 1/2 case, it's assumed that answering the same question correctly twice isn't inherently better than answering it correctly once - for example, you have to debate an opponent at each awakening. You're either right or wrong and you don't get extra credit for winning the same debate with the same opponent twice.
Implicit in any answer is a way of valuing the choices made. In the 1/3 case, it is assumed that being right twice is better than being right once - for example, you get $100 each time you answer correctly. In the 1/2 case, it's assumed that answering the same question correctly twice isn't inherently better than answering it correctly once - for example, you have to debate an opponent at each awakening. You're either right or wrong and you don't get extra credit for winning the same debate with the same opponent twice.