I thought there were a latest EU MIUI 12 Stable version, but no and your explanations are crystal clear.I still don't get your question. You already have the latest EU MIUI 12 Stable version available.
You get the update notification because there is a MIUI 13 version available. You can't install that through recovery.
Even if you could install it through recovery, you'd still be forced to do a clean install because you are switching Android versions (11 to 12), thus losing data and having to deal with app and personal data backups, etc.
Gonna prepare for a clean install then