I think a reasonable interpretation of the colloquial sense of incompatibilist free will is that people want to be (or have the experience that they are) their own causal origins or prime movers. That they originate an action that is not (purely) the effect of all other actions that have occurred, but in such a way that they decided what that action was.
From the outside, this is indistinguishable from randomness. But from the inside, the difference is that the individual had a say in what the action would be.
Where this tends to get tangled up with notions of a soul, I think, is that one could argue that such a free choice needs some kind of internal state. If not, then the grounds by which the person makes the choice is a combination of something that is fixed and their environment, which then intuitively seems to reduce the free-will process to a combination of determined and random. So the natural thing to do is then to assign the required "being-ness" (or internal state if you will) to a soul.
But there may exist subtle philosophical arguments that sidestep this dilemma. I am not a philosopher: this is just my impression of what commonsense notions of free will mean.
My point is that from the outside this doesn’t look like randomness at all, unless you are mistaking ignorance of their motives as a random oracle. If you can infer what set of goals drives their decision making, and the decision making process itself (e.g. ADHD brain vs careful considered action) you can very much predict their decisions. Marketing and PR people do this every single day. People don’t behave like random oracles, they behave like deterministic decision makers with complex, partly unknown goals so our predictions of their behavior are not always correct. That’s not the same thing as random.
From the outside, this is indistinguishable from randomness. But from the inside, the difference is that the individual had a say in what the action would be.
Where this tends to get tangled up with notions of a soul, I think, is that one could argue that such a free choice needs some kind of internal state. If not, then the grounds by which the person makes the choice is a combination of something that is fixed and their environment, which then intuitively seems to reduce the free-will process to a combination of determined and random. So the natural thing to do is then to assign the required "being-ness" (or internal state if you will) to a soul.
But there may exist subtle philosophical arguments that sidestep this dilemma. I am not a philosopher: this is just my impression of what commonsense notions of free will mean.