Could you summarize the evidence that the self-reported perception of the nature of willpower was the primary difference between the two groups, as opposed to both groups accurately reporting the qualities of their own willpower?

Without strong evidence in this regard, the finding is trivially explained by the well-known finding that you mentioned earlier: better impulse control is generally correlated with positive outcomes; if some people actually have abundant willpower, then they presumably would do better at tasks that require willpower. All very simple, and not much influenced by what theory of mental capacity people may or may not hold.

So the evidence of different perception, not capacity, is key here.