I have discovered that I can change the name in the menu from “Language Switcher” to “German” or “English” if I go to Customize > Menus. However, this changes the button for both the current language and the other language to the same thing (either both say “German” or both say “English”.
To answer your question, if I also enable it for Deutsch, then the English menu appears on both the English and German version of the website and the German menu never appears. I’ve attached some photos below to try and show you.
I have two menus, one in English -Menu 1 (Primary Menu English)- and one in German -Deutsch (Primary Menu Deutsch)
Menu 1 (Primary Menu English) has the display location set as “Primary Menu English”
Deutsch (Primary Menu Deutsch) has the display location set as “Primary Menu English” (Note: I included some of the menu items above so you can see if I’m looking at the English or the German Menu)
If I also enable Menu 1 (Primary Menu English) for Deutsch as you suggest then it automatically disables Deutsch (Primary Menu Deutsch) for Deutsch
On the front end the only effect this seems to have is that the English menu is permanently present on both the English and the German versions of the site.
As you can see, there’s still no flags, and the words “German” show up twice, instead of “English” and "German.
Would it hep if I gave you Editor abilities so you can see the back end?
Thanks in advance!