How can I undo "add name to this face" in Photos?

I find nothing in the documentation about that feature of people that allows you to "add a name" to a person. I'm not sure what they were thinking when they made that option. Nicknames maybe?

Have you seen this page in the Photos User Guide? It is not very detailed, but covers the basics.

https://support.apple.com/guide/photos/view-photos-by-whos-in-them-phtad9d981ab/ mac

When you click into the name field at the top of the album for a person, you can pick a suggested name from the drop down menu that appears - usually the names you already used to name people, or the names from your Contact cards. But you are free to use any name you want to use. If you select a name from the drop down menu, the person will be added to the existing album with the name you picked. This way you can merge the albums for two persons. It is easy to merge them by mistake, but a lot of work to separate them again.

When I select of of "John's" pictures and manually change them from Angela to John and then let the program do it's magic... nothing happens

That should work. But you have to change all of John's faces manually back from Angela to John. Photos will not split the merged album again automatically.

It may be easier and quicker to restore the Photos library from your backup. If you are backing up with Time Machine, restore a version of the library from before you merged the two persons.