How did the Carolingian Emperors and Holy Roman Emperors refer to the Byzantine Emperor and his Empire?

by No_Longer_Lovin_It

I have heard that using "Byzantine" to denote the Eastern Roman Empire is a modern convention, so what terms did the pre-modern claimants to the Roman Title use to describe the commonly called "Romans."

WelfOnTheShelf

Much more can still be said about this, especially from the perspective of the Carolingians (who are a bit early for me), but I've written a few previous answers that might help:

What did the other people's of Europe call the Byzantine Empire? Did it differ depending on who you asked and were there any changes as the empire dwindled?

What was the Eastern Roman reaction to Charlemagne being crowned Roman Emperor by the pope in 800?

In 1185, Holy Roman Emperor Frederick Barbarossa arrived at Constantinople. Do we know anything of any meeting with the reigning Byzantine Emperor, Isaac II?

In short, when Charlemagne was crowned emperor everyone seemed to think there were two Roman emperors again, just like there had been in the past, so it wasn't much of a problem. But the later Holy Roman Emperors called the Byzantine Emperor "the Greek emperor" and the Byzantine Emperor called the HRE "the German emperor" and eventually they both felt extremely insulted.