I find the definition from Searle very convincing.

The thesis of determinism asserts that all actions are preceded by sufficient causal conditions that determine them. The thesis of free will asserts that some actions are not preceded by sufficient causal conditions. Free will so defined is the negation of determinism. (Searle 2007: 47)

From the graph and the definition, we can see that Searle does not hold a compatiblist approach. He just says that we cannot use the form “A caused B” (which is for third-person ontology) to describe the first-person conscious experience, which is in the form “a rational self S performed act A, and in performing A, S acted on reason R).

In the language of neurobiology, the formulation of free will is like this:
If the state of his brain at t1 is not causally sufficient to determine the subsequent states of his brain up to t2, then, given certain assumption about consciousness that I need to make clear, he does have free will. (Searle 2007: 61).

More precisely, free will is a phenomenon in time, if it exists at all. It means that “at any instant the total state of consciousness is fixed by the behavior of the neurons, but from one instant to the next the total state of the system is not causally sufficient to determine the next state”. (Searle 2007:65)

Searle purposes two hypotheses. The first hypothesis is epiphenomenalism. It fits our neurobiological knowledge. However, it is unattractive as it believes that our conception of having free will is illusionary. The second hypothesis is more complex. From premise 1 that all indeterminism in nature is quantum indeterminism and premise 2 that consciousness is a feature of nature that manifests indeterminism, we have the conclusion that consciousness manifests quantum indeterminism. This means the “gaps” in the above graph is filled by quantum randomness. However, randomness at micro level does not by itself imply randomness at the system level. It seems clear enough.

After his formulation, the problem of free will is largely reduced to something that leaves for the work of neuroscientist. I believe that his theory is another form of physicalism.