Yes, "source available" does mean exactly what it says. There is a class of projects, however, which are still looking for a term. They are all "source available", but that's not all there is to it (you could call all FOSS projects that too, but nobody does). The main difference is that they allow you to modify the code and even share the modifications ("source available" projects generally don't).

Arguably, "open source" is the correct term to use, and FOSS should be called "free source", but OSI made a mess there.

"Fair source" is as good a term as any. "Cloud protected source" (as in "cloud protection licenses") also.

Current situation is not good for anyone except BigTech, but sure, let's burn anyone trying to avoid unfair competition by actually using a "fair source" license.