You can just check SEP:

https://plato.stanford.edu/entries/compatibilism/

https://plato.stanford.edu/entries/compatibilism/supplement.html

They have different specific accounts for compatibilism -- example higher-order theories of freedom (from Frankfurt and others), Reason-responsiveness views, and there are also compatibilist variants of "ability to do otherwise".

Also compatibilists are trying to make many different points:

• Some may argue that what we actually want to "track" by freedom and what we care about are compatible with determinism. This can involve some thought experiments and arguments as to how incompatibilist "ability to do otherwise" doesn't really offer anything much.

• They may argue that "ability to do otherwise" itself is compatible with determinism if ability is understood in a unloaded/unbloated sense.

• They often want to argue not only that we have compatibilist free will, it's also moral responsibility inducing. Which is a substantive point and not just "shrugging".

• They may attack incompatibilist intuitions for example -- they may provide cases where it feels intuitive to assign praise even when the person says they are compelled by their nature to do some good, or they may argue lack of meta-wants or meta-meta-wills to control oneself and such are unnecessary demands and not clear why necessary for moral responsibility. And so on.

• They may also provide x-phi support that ordinary humans have elements of compatibilists intuitions.

> This sort of solution essentially splits freedom into two concepts: the type of freedom we recognize in everyday life, and freedom from the laws of causality. Since the latter is impossible, it makes no sense to draw any kind of moral consequence from it, and one must therefore focus on the former. This is rather unsatisfying because it feels like the philosophical version of a shoulder shrug.

But that sounds more favorable to compatibilism than against. If the compatibilist's version of freedom is the very freedom we recognize and talk about in everyday life, what's the practical value and meaning of this "freedom from laws of causality" (which you yourself recognize to be ultimately seemingly incoherent, because to be free from causation is make actions free from the actor which would again be no freedom at all)? So why should anyone bat an eye or lament or celebrate the non-existence of some concept that cannot be even legibly conceived of? It's also not clear if moral responsibility is necessarily threatened by lack of such "freedom from causality". Backward-looking punishment can also be independently argued against. So we don't have to worry about that.

Personally, I am not a compatibilist. I am just trying to give credit where it's due.