"Source available" is definitely in much wider use than "weights available" but it's not that common and it's not used consistently at all in my experience.
Weights are different enough from source code I don't find the metaphor useful or compelling. I don't see any value in trying to apply it to a novel subdomain of software engineering, frankly.
Anybody calling SQLite "source available" is mistaken. Are you sure they weren't confusing "source available" with "open core" (Open Source + proprietary add-ons)? SQLite's source code is public domain, which meets the Open Source Definition and is true Open Source. But they also sell proprietary add-ons to fund its development:
No one calls SQLite "source available", and if they do, they're doing it more as a protest of their development model than being serious.
https://gitlab.com/gitlab-org/gitlab/-/blob/master/ee/LICENS...