There is actually rather little published research on all this, so you could get a handle on the literature quite quickly if you go and read the references in the thread. Read the whole thread for yourself, by all means, and better still read the articles and other references that are cited, mainly by Levy. Don't take my word for anything.
But ... meanwhile, one thing that leapt out at me was this part of Levy's discussion of the empirical research:
In response [to an earlier article], Nahmias et al. 2007 (Free Will, Moral Responsibility, and Mechanism: Experiments on Folk Intuitions. Midwest Studies in Philosophy 31) argue that the description of determinism Nichols and Knobes used causes people to think that future actions had to happen no matter what anyone did. That is, the descriptions suggested fatalism, not determinism (where an event is fated if I can’t avoid it, like Oedipus’s killing his father – it doesn’t matter what Oedipus does, because the event occurs independently of what he does). Nahmias et al. therefore suggests that it is the incompatibilist response, and not the compatibilist response that is a performance error. They think that incompatibilist intuitions are caused by confusing determinism with bypassing mental states: people are not worried that their thoughts are determined; they are worried that their actions might be determined regardless of what they think. In support of this, they showed that if determinism was described in psychological terms (‘once speciﬁc thoughts, desires, and plans occur in the person’s mind, they will deﬁnitely cause the person to make the speciﬁc decision he or she makes’), incompatibilist responses fall dramatically.Nahmias et al might be wrong, of course, and the earlier study in which the folk appear to reject compatibilism might be right. But in any event, if we want to get a grip on what is actually bugging the folk when they worry about free will, we'd do well to see what understandings of their understandings are defensible on the basis of the empirical research. It does seem from this description that what is bugging the folk may be the question of whether fatalism is true. If so, "free will" for the folk isn't anything as spooky as often asserted by Jerry Coyne and Sam Harris, but, in essence, the denial of fatalism.
Now, if all that is meant by "you have free will" is that certain fatalist (and/or perhaps passivist) views are false, compatibilism is in good shape. However, the concept of free will may not have a lot to contribute in, e.g., theodical debate. If so, so be it (I never actually thought it did). In any event, I am not arguing here in any settled, dogmatic way that that is all free will means to the folk.
Rather, it seems to be an open question what free will means to them, needing further empirical research - though there seems to be some fairly strong support for the idea that "acting freely" (not necessarily, in my view, the same concept as that of having free will) has a fairly stable meaning for the folk. Namely: "they described free actions and choices as those free from constraints like compulsion and coercion and regarding which they had time to deliberate."
If that is all free will means to the folk (i.e. if they conflate the concepts of free will and acting freely), then it does appear that the folk view of the matter is (a) not spooky, related to religion, or related to dualistic notions of the mind; and (b) not likely to be useful for theodical purposes (theologians might as well give up on the idea).
There's more to be absorbed in all this, but these studies appear to be grist to my mill that the folk don't necessarily have a spooky notion of free will ... and that books, articles, and blog posts which seek to refute some spooky notion - then conclude that we don't have free will, as "free will" is usually understood! - are on shaky ground.