Perhaps a very provisional form of freedom....
And that, I think, is essentially the way many compatibilists use the word free in free will.
Except that will has no autonomy of decision making or action, will does whatever the brain is doing....just as it is not the legs that act of their own accord but are moved by impulses sent by the brain, the provisional, narrow usage of the word does not prove the proposition. Provisional 'freedom' only refers to an aspect, a condition, a given ability....the dog is free from its chain but is constrained by the fenced enclosure, the dog is not free to roam being references to states within the system, not nature of the system as a whole.
The principle on which compatibilism rests does not actually make it compatible with determinism because determinism does not allow one to do otherwise, within a deterministic system, freedom is an illusion.
Yes, but a leg isn't 'really' free to move either, so all I'm saying is that the word 'free' in free will can be validly used, in a provisional, limited or colloquial way, imo.
I can still say, if I want to, which I sometimes do, that at the end of the day it's the wrong word (for reasons given) and you can say that it cannot (reasonably) be used. Your saying that isn't arbitrary or irrational, nor is it a fallacy, but it is just your reasoned view, and strictly-speaking wrong in the absolute (non-subjective) sense, since the word free can (reasonably) be used, via different reasoning. Your view (and perhaps mine) might be special pleading in an informal sort of way. Or at least let me say I get that general objection. I think we should call them our reasoned opinions regarding labelling, not an absolute decree.
Moving on (hopefully), I pretty much agree with almost everything you say.
There is, looming on the horizon, for everyone, the prospect that everything we think or do is determined and/or randomly caused and that beyond that we cannot think or do anything otherwise than what we did think or do in any given instant. The 'algorithms' in our brains would, it seems, have to churn out exactly the same output if the system were re-run (allowing for the possibility of randomness).
At least it seems very difficult indeed to make a case for there being a convincing alternative to this hypothetical.
This would mean that if a certain person commits a crime, that person literally could not have done otherwise and we would have done the same had we literally been in that person's shoes, so to speak. Of course, we would be them in that case, obviously.
In that scenario, compatibilist free will would be 'fully trapped anyway' and we might ask, in what way is it reasonable to hold that person morally responsible for what they did? I don't mean to suggest there isn't an answer to that. I have read several. But this is what I mean about tricky questions which follow on after the labelling issue has been set aside.