Agree with your main point, but Lingua Franca doesn't refer to French but to "Mediterranean Lingua Franca", an actual trade language of its own which was a mix of a bunch of things. It had that name because it was "the language of the Frankish" (traders) but what they (eastern sea/Muslim traders) meant by "Franks" was any western European.

https://en.wikipedia.org/wiki/Mediterranean_Lingua_Franca