The main thing I'm disagreeing with is undefining what CAP means. CAP is a very particular proof and the rest I think perhaps we need a new name for.
> CAP has a long history and was originally provided without a proof.
Irrelevant. All of database history is filled with what today is comparatively junk. The understanding has improved and we have better ways of talking about the particular subject matter.
> The proved version of CAP has a very strict definition, but it's not that useful in practice because most systems don't actually follow that model.
It is useful in what it proves, so don't bend the definition. This is my main point. You will have to invent another taxonomy perhaps. We have good names for all the typical consistency models I believe, so we can use those as a starter.
> Most notably, the CAP theorem is often still true when "consistency" is defined as weaker than "linearizable".
This is the type of reasoning that just muddies rather than clarifies.