I have a felling that you are combining two points into one here:
1). "AI art" is not liguistically precise, so we should use "AI image generator" instead
AND
2). Artificially generated art should not be consider "art".
When it comes to 1) - well, it's probably true that "AI image generator" is more precise than "AI art", it is still not precise at all. As an AI engineer I know that the term "artifical intelligence" has nothing to do with any intelligence - it's just really sophiscticated math. So shouldn't we try to force the name "sophiscticated math applied as an image generator" instead? It's much more precise (but probably still not precise enough).
Also - for people who doesn't know what AI is, I think this is a pretty accurate name actually. Images, paintings, etc. are considered art. They are generated by some magic thing called "AI". And we have a pretty descriptive name "AI Art" - is languange more about being scientifically precise or rather about being descriptive?
The 2). pretty much comes down to a very subjective and individual question of "what is art?". And I believe that other commenters had made a clear argumnets for why the whole process of designing and creating such AI generated images CAN be considered "art" and why for some people AI generated images CAN be viewed as "art". But it's purely subjective at the end of the day.
My point is - languange is not about being precise and art is what you consider it to be. It's impossible to have a precise name for everything, especially something as not-easily-definable as a genre of art.
And while I think I may agree with your last sentence

"AI image generator” is more precise and doesn’t lose anything except unjustified hype

it's not a good reason for me to enforce thousands of people to use different name.