How to tell git status to report that the checkouted tag has changed?

The "detached at" and "detached from" messages rely on clues that git checkout leaves behind in the HEAD reflog. They are not reliable and you must not rely on them. But if you want git status to provide the information you're looking for more often, you can get that: simply check out (git switch or git checkout) some named branch between the check-out operations on tags. This will update the HEAD reflog in such a way that the clues are there more often.