This is something I just don't understand, a lot of times I read about Jerusalem falling into Frankish hands during the Crusades or how Saladin ended the Frankish occupation of Jerusalem and more.
But this doesn't make any sense whatsoever....
The Frankish empire had been divided at that point for a long time, the HRE was even already set up. Maybe France was still referenced as the Frankish empire but it still wouldn't make much sense because the kingdom of Jerusalem, which Saladin defeated, was created by English Normans who have no connection to the Frankish empire...
I mean, I see people call all crusaders "Franks" but only a small portion of the crusaders came from France and the Frankish empire didn't even exist anymore. It's like calling Italians Romans...
So, why is the word "Franks" used so much? Is there a reason for this or is it just plain wrong, as I assume?
It mainly comes from, as you pointed out, the Arabic sources, for whom the word was a catch-all term for Western Europeans. It was used by the crusaders as well however, most famously by the author of the famous First Crusade narrative the Gesta Francorum (lit. 'The deeds of the Franks'), despite the fact that the author was (probably) from Norman Southern Italy.
It does make sense as well because, despite your assertions, the majority of Crusaders due seem to have come from modern-day France. I can't link it on mobile, but if you check my post history you can find a comment I made a few weeks back on the diversity of the Crusading armies, but most of the manpower, certainly in the first century of crusades, came from France. Finally, the Kingdom of Jerusalem was not founded by English Normans, but rather by Lotharingians, from the German empire. After the death of Baldwin I, the kingdom passed to a French dynasty. In fact, I'm not aware of any documented English involvement in the First Crusade, but there was certainly no major role for the English until Richard I and the Third Crusade.