The misunderstanding seems to be that "certificate" and "solution"/"answer" are the same thing. Formally, they are not. Intuitively, they can but don't have to be.

@Raphael In a sense, they have to be different, since NP is a class of decision problems (e.g., "Is this formula satisfiable?") so the "answer" is always just "yes" or "no". Unless the problem is in P, "yes" or "no" isn't going to work as a certificate, since it can't be verified. But, with a more general notion of "answer" (which is surely what you meant), I completely agree.

@DavidRicherby That's exactly what I meant, yes. :) Practically, "yes" and "no" is not the answer to the problem; decision problems are merely easier to investigate formally.

That said, it should technically be possible for the "answer" to be exponentially large even though the decision problem is in NP. That would be a good teaching example to exhibit the difference; such an "answer" could not be a certificate!