Can I edit the tag on the community question? Like, change its name from one to another?
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No, you have to add a new tag and remove the old one, if you're looking at a single post.
Moderators have a function where we can change them en-masse, but it's still not an "edit", it's a "search and replace"
I was watching this question and even have a screen of it where there is a "approval" tag on it. But today there is the "approver" tag.Is this the case where the moderator removes one and adds another?
Possibly, but other people can add tags as well.
It won't have been a moderator simplifying/merging tags - they only do it for spelling mistakes and consistency. Neither approval nor approver are candidates for those (see https://community.atlassian.com/t5/tag/approver/tg-p and https://community.atlassian.com/t5/tag/approval/tg-p )
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