You’re exactly right that this varies over time, but also by the source and the author’s choices when writing it. For example, a lot of the narrative writers used archaizing language at times, while not necessarily representing what they actually called the place. Also, in a politically fragmented locale like Western Europe, states as we now conceive of them did not exist, so instead of calling a place ‘France,’ Western medieval writers usually spoke about the people who lived there: e.g. the Franks, or more specifically, the Burgundians, the Aquitainians, the Normans, etc. as opposed to France, Burgundy, Aquitaine, and Normandy, respectfully (though this is not always true).

For the purposes of your question, this means that many sources would not necessarily call the Byzantines the “Roman Empire” but “the Romans,” which was probably the most used term. Especially later, like around Crusader times where there was more western contact with the Byzantines, the term “the Greeks” came into fashion too (likely in reference to the fact that the Byzantines spoke Greek), but “the Romans” is probably the most common.

As far as the Papacy specifically, I think I’m less qualified to answer, but my memory of the papal correspondence surrounding the schism of 1054 suggests that the Emperor was called “Emperor of the Romans.”

There is no evidence that Constantinople was ever called “Rome.” The appellation “New Rome” in Constantinian times doesn’t seem to have stuck and only appears later when historians are being particularly flowery in their language. The calling of the Byzantines “The Roman Empire” or “the Romans” has little to do with the name of the city of Rome, and everything to do with the political continuity of the Byzantine state from antiquity up until whenever the author was writing. This is exemplified by the fact that in the aftermath of the Seljuk conquest of Anatolia, which was previously the Byzantine heartland, a name for one of the splinter kingdoms was “The Sultanate of Rum” (i.e. Rome), referring not to the city, but to the fact that it was a part of the politically continuity that was the Roman Empire even in the 12th century. Parts of the Balkans were called Rum by the Ottomans after their conquests for similar reasons.

The names”Byzantine,” “the Byzantines,” or “the Byzantine Empire” doesn’t seem to have ever been used to refer to this place during the Medieval Period. It was probably first used in the 16th century, in a collection of Byzantine texts compiled by Hieronymus Wolf, and then proliferated in academic sources ever since. Constantinople, however, was often archaized as “Byzantium” (e.g. Nicetas Choniates, 12th-13th c. author), but that was mostly from the Byzantine side.

Apologies for any confusion I may have made by calling them “the Byzantines.” Personally, I think it’s reasonable to call them that in order to highlight the vast cultural, social, economic, linguistic, and even political changes that separate the vast, multiethnic Roman Empire of the 1st-5th/6th/7th centuries to that of the 10th-15th centuries, as long as you recognize that it’s a later, academic appellation.