I feel like most of these schemes to categorise data as "public but not really" are ultimately doomed to failure. Even if you could trust every AI company in the world to respect these terms, is there anything stopping someone else indexing the data and selling them the information? I know there's copyright law but they're apparently ignoring that anyway.

Ultimately this reminds me of those really early social media profiles (before people understood privacy settings if they even existed) which would say "If you're not my friend you're not allowed to read this page".

If you don't want your content to end up in some database/archive don't publish it for the whole world to see.