Question: Assume A is in fact right. Is there a difference between my doing A because A is right, and my doing A because I think A is right?
Answer: If that A is right (call this reason R) and that I think A is right (call it TR) are distinct reasons, then there is a difference between doing A because of R and doing A because of TR. But R and TR are distinct reasons. An easy way to see the distinction is that there are actions that TR, when it obtains, justifies directly which R, when it obtains, does not directly justify. Specifically, TR, whenever it obtains, directly justifies me in asserting "I think A is right", while R, even when it obtains, does not directly justify me in asserting "I think A is right." After all, R might obtain without my knowing about it, in which case I would not be justified in asserting "I think A is right" (if I asserted that, I would be lying). And even if I know that R obtains, it is not R that justifies my assertion "I think A is right", but it is TR. If two reasons directly justify different actions, the two reasons must be distinct.
Remark: There is something wimpy about thinking of oneself as doing A because one thinks (or even knows) that A is right. Surely, under normal circumstances, the reason one thinks of oneself as having is not that one thinks that A is right, but that A is right. This is, of course, a standard point.