Why do I have two notes applications on my iPhone Home Screen ?

It is possible to duplicate the app icons on the Home Screen. This is most likely your case. Long-press one of them and choose "Remove from Home Screen."

Aaand now my wife’s notes are all gone. iCloud was full and not backing up. This what happens when you delete that extra mysterious Notes icon.

Thanks for your input.

Now, before I'm deleting the 2nd Notes app (or removing from the Home Screen) I'm making sure that I have it covered with backup(s)!