You might as well call me English too (I live in Wessex after all)!
What WP seems to have forgotten to mention is who actually used those terms. I'm prepared to be shot down but I doubt there was a properly defined concept of "Western European" between say 300 to 1500 in the right hand side bit of what we now call Europe.
What you are on about is the left and right Roman Empires (my terms) and how they talked about each other.
The left lot is now nominally modern Germany and France and much besides, what was called the Holy Roman Empire. The right lot is the Byzantine Empire.
Now, your Franks were part of the left lot. The Franks were a tribal identity, cf Saxon, Angle, etc. The Franks is where la belle France gets its name (cf Frankreich, and Französisch - German names for France and French.)
I am English (by assertion and routine) but my family name says I am descended from Germans on the male line (Brits being patrilineal.) As it turns out 300 odd years ago from a Hanoverian merchant seaman and (very) loosely on my matrilineal line 500 odd years ago I have a ancestor from Padstow in Cornwall. I also have inroads from all of the home nations (England, Ireland (both), Scotland and Wales). There are also quite a few other inputs to my blood line from much further abroad.
One of my uncles is quite fastidious in his research and has many thousands of people in the DB.
https://en.wikipedia.org/wiki/Mediterranean_Lingua_Franca#Hi...