Pamela Hieronymi, in 'The Wrong Kind of Reason', distinguishes between "constitutive" and "extrinsic" reasons for an attitude. The difference is that they address different sorts of questions. Roughly put, constitutive reasons speak to whether the content of the attitude warrants taking the attitude towards it. So, in case of beliefs, these would be reasons or evidence that the thing believed is true. Extrinsic reasons, by contrast, speak to the question of whether the attitude itself is a good one to have, independently of the merit of its content. So, if I offer you a million dollars to believe that grass is blue, then that's an extrinsic reason for the belief.
Constitutive reasons for believing p are then reasons that bear on the question whether p. They are truth-indicative reasons. Further, as Hieronymi notes, "Finding them convincing and so settling the question on which they bear amounts to believing." This ties in with the subjective evidence principle and the phenomenon of deliberative transparency that I was discussing the other day. Constitutive reasons bear on a question the settling of which amounts to the belief. Extrinsic reasons are different. They bear on a different question, the question whether it would be good to believe p. (Settling this question amounts to a second-order belief, that is, a belief about this question, a belief about whether it would be good to believe p. But they do not constitute a belief whether p.)
This seems a helpful distinction, and one that illuminates Musgrave's distinction between reasons for belief and reasons for the thing believed. In particular, extrinsic reasons are reasons for believing, but not reasons for the truth of the thing believed.
Now, Hieronymi calls these the "wrong kind" of reasons, but I'm not sure what is meant by this. Perhaps it's simply a way to highlight that they are a different kind of reason, and not one that speaks to the truth of the belief (as we might normally expect reasons to do). I don't see what more it could mean. It does seem undeniable that extrinsic reasons do exist and that they genuinely are reasons. Surely evidentialists cannot deny this. So what are they denying when they claim that truth-indicative reasons are the only reasons for belief?
Evidentialists must say that extrinsic reasons aren't reasons for believing per se, but rather are reasons to get the belief. But what substance is there to this distinction? It seems like a mere semantic quibble to me. Everyone agrees that there are constitutive and extrinsic reasons for belief, and we merely disagree about how to describe them. Anti-evidentialists want to call both kinds "reasons for belief" (whilst still recognizing, of course, that they are very different kinds of reasons). Evidentialists want to give extrinsic reasons a different name, instead calling them "reasons for getting the belief". But what's in a name?
The important thing is that there are non-evidential considerations that can count in favour of beliefs. If clapping my hands would save the world, then this is a reason to ("get" myself to) clap my hands. In exactly the same way, if believing that p would save the world, then this is a reason to believe p. It's what you ought to do, no matter the truth of p. You should take whatever means you can to achieve this end. None of this can plausibly be denied. So what in the world is this debate about!?
Update: retraction here; see also comments.