Compatibilism is simply the belief that free will and determinism are compatible. ... Compatibilism makes no claims about determinism/indeterminism.
If compatibilism makes no claims about determinism, then it is not possible for compatibilism to maintain that free will and determinism are compatible. At the very least, compatibilism most certainly has to characterize - make claims about - determinism. You might mean that compatibilism need not hold determinism to be a correct characterization of actuality, but that has not been the context of the ongoing discussion thus far. So, let's take up this different context.
Compatibilists can be determinists, indeterminists or agnostic
This suggests that compatibilists are to be characterized best, first, and foremost as analysts of the possible relationships between determinism and free will. That would certainly be the case for the agnostic compatibilists.
Such analysis can begin by addressing the notion of determinism. You have not (yet) objected to the characterization of determinism as, in effect, denying that there is macrophysical indeterminateness. The analysis then addresses the notion of free will. The notion of free will can be considered as a matter or description of human experience.
On occasion, there are humans who think they have the experience of there being actual indeterminateness. That experience is not of there being limitless possibilities available, but it is common for humans to think - to have the sense - that the indeterminateness is what affords a freedom to settle the apparent indeterminateness. Of course, humans in such a circumstance do not think that actuality stops to await their decisions or choices. It is as if actuality has its own momentum. In which case, the perceived indeterminateness is akin to an opportunity to interrupt the momentum of actuality or to change the course which actuality would otherwise take.
Based on this experience, indeterminateness is necessary for there to be freedom; indeterminateness is necessary in order for a person to be free; indeterminateness is necessary in order for a person to be free-to
actually decide or choose.
Of course, the compatibilist analyst would here say that a determinism which denies that there is actual macrophysical indeterminateness is a determinism which is incompatible with there being the sort of freedom for which indeterminateness is necessary.
This means that a compatibilist who is not just an analyst but is in fact a determinist is a compatibilist who is an incompatibilist with regards to the sort of freedom for which indeterminateness is necessary.
And, yet, as the compatibilist analyst would note, the compatibilist-determinist is still claiming that there is a sort of freedom for which indeterminateness is not necessary.
Typically, this freedom is described as the state of being free-from external coercion or control, and that being free-from is as much a matter of human experience as is the experience of there being apparent indeterminateness. Indeed, in addition to indeterminateness, even this sort of freedom - this being free-from - is necessary in order for a person to be free-to in the manner above discussed. It is just that, from the perspective of there being actual macrophysical indeterminateness, being free-from is not sufficient for being free-to. And this is to say that a compatibilist-determinism which denies that there is actual macrophysical indeterminateness is incompatible with that freedom for which indeterminateness is necessary for being free-to. Therefore, this sort of compatibilism (this compatibilist-determinism) is an incompatibilism, and a person who holds to this sort of compatibilist-determinism is a compatibilist who is an incompatibilist.
The compatibilist-mere-analyst would then take up the case of the incompatibilist-determinist. The incompatibilist-determinist does not - or need not - deny the experience of being free-from external coercion or control. Rather, the incompatibilist-determinist can simply hold that being free-from is not a sufficient description of being free. Accordingly, the incompatibilist-determinist objection to compatibilist-determinism amounts to a charge of there being a failure to disambiguate the term
free.
Or, in the alternative, the incompatibilist-determinist might more aggressively insist that compatibilist-determinism relies on an intentionally ambiguous and non-necessary way of using
free. This more aggressive charge just highlights that the disagreement between the incompatibilist-determinists and the compatibilist-determinists boils down to an argument over usage of the word
free.
But no manner of expression is to be regarded as actually necessary. All manners of expression can be presented alternatively, and, in this case, that would mean without use of the word
free in any of its forms. Both the incompatibilist-determinists and the compatibilist-determinists seem to deny that there is actual macrophysical indeterminateness. In this regard, both the incompatibilist-determinists and the compatibilist-determinists are incompatiblists. It also seems that both the incompatibilist-determinists and the compatibilist-determinists can agree to there being the experience of there being occasions which are without external coercion and control.
Of course, the foregoing is in a physicalism context, and the incompatibilist-determinist and the compatibilist-determinist positions discussed have been from the perspective of a non-reductive physicalism. An ardently held reductive physicalism viewpoint could introduce other possibly relevant factors that would lead to disagreements beyond that related to use of the term
free.
The bottom line is that even if compatibilism is not necessarily a determinism (which is to say if a compatibilist-mere-analyst is agnostic with regards to whether determinism corresponds with actuality), compatibilist-determinism is an incompatibilism, and compatibilist-determinists are incompatibilists if they deny that the experienced macrophysical indeterminateness is or can be actual macrophysical indeterminateness.