The Byzantines always called themselves 'Roman' and had no real reason not to see themselves as a continuation of the earlier Empire. Constantine wasn't the first to shift the center of government away from Rome itself, military and legal continuity was maintained, and it was a couple of centuries after him that Byzantium really took on any 'Greek' character. A classic example of how consistent they were in this terminology is the Turkic Sultanate of Rum was so named because they thought they had conquered the land of 'Rome' by ruling central Anatolia, not because they made the best Mai Tais in the land.
The dispute of which emperor is 'Roman' has its roots in a very poorly-timed difference of succession laws. In 797, Irene (who had been ruling as regent for her son for nearly two decades) deposed her son, who died in the process (I don't know if there is a modern consensus on whether she had her son blinded in a way that was meant to kill him, but contemporaries certainly thought so). She then reigned as Empress--basilissa, not basileus (with maybe a couple exceptions, but the point is she was ruling as a woman and everyone knew it). There's already been strife over iconoclasm, brutally murdering your son is rarely a good look, and the empire was still withering under the assaults from the east--during her time as regent Irene was forced to pay large sums to Harun al-Rashid (yes, 1001 Nights Harun al-Rashid) to obtain peace, since she couldn't win on the field, and had commanders repeatedly defect. So, its maybe not so good to be king, its already an unstable time, and Irene being a woman isn't going to win her many extra friends.
At the same time, Charlemagne was making waves (not just on The Breakfast Club), having taken the Lombard lands in Italynearly a decade prior, and having just gone east and crushed Saxony and Pannonia. He was now preeminent in the west, from northeastern Spain to Denmark to Hungary to central Italy. Now, Frankish Salic law is pretty explicit in only allowing males to inherit, thus disallowing female rulership. So, from the point of view of the Franks, there is no Roman Emperor. Whatever they may have thought about the 'geopolitical' ramifications, there should be no doubt that they thought they were legally in the right. So, in 800 the pope crowns Charlemagne as 'Emperor of the Romans'. Irene is pissed.
However, not much immediately comes of it. The Byzantines are still relatively weak from internal strife and eastern defeats, and Charlemagne doesn't seem to have had much desire to push further east. The immediate reaction is mostly annoyance and trying to agree on the finer points of the border/sphere of influence. There had already been fighting between Charlemagne and Irene when he invaded Italy due to competeing claims between the Franks, Lombard, and Byzantines in the areas of Ravenna and Benevento. Around that same time, Irene had broken off a betrothal between her son and Charlemagne's daughter, but I don't know whether that was cause, effect, or unrelated. There is fighting over Venice that lasts until nearly a decade after Irene's death, but I can't tell if that started during her reign (either way, internal Venetian politics between pro-Frank and pro-Byzantine factions is more the cause than direct empire-to-empire hostility).
Irene is deposed in 802, so she has doesn't get time to do much more than resent Charlemagne's claim. Her successor Nikephorus I does indeed register the formal disagreement with the Frankish usurpation of his title when he negotiates with Charlemagne over who gets what parts of northeast Italy. Wars with the Bulgarians consume his attentions and eventually his life, so essentially the three rulers Irene, Nikephorus, and Michael I had some minor skirmishes with the Franks, conceded control of Lombard Italy but not the imperial title, then went back to getting killed (the years 800-820 saw five Emperors, only the last of whom died naturally in office). By the time of Leo V, both empires were focused on internal matters, witht he Byzantines having another go at iconoclasm, and the Franks under Louis the Pious trying to be Franks not under Louis the Pious (I think 3 or 4 civil wars over 20 years). So, the issue got tabled, and it wasn't until the Ottonians a century and a half later that both claimants to the title were powerful enough to write strongly-worded letters to each other.
A letter about 400 years later, between Isaac Angelos and Frederick Barbarossa, demonstrates the difference in nomenclature. The 12th century letter from Isaac calls himself 'Emperor of the Romans' and calls Frederick 'King of the Germans' and was taken as quite insulting by the recipients. A earlier letter from Italy (a very different, much more direct, rivalry at the time than the later HRE ever was) shows Louis II defending his usage of the title 'Emperor of the Romans' while the Byzantines appear to have called him 'Emperor of Frankia'. Louis calls the Byzantine Emperor 'Emperor of New Rome' and his basic argument is that the title 'basileus' in Greek-language histories isn't limited to the most prominent ruler of the Mediterranean world, but also given for rulers of the Goths to the Egyptians, so lets just name ourselves after a city we rule and that's totally equitable. Pretty slick, and a lot more diplomatic than just calling them the 'Kings of the Greeks' (which Pope John XIII did about 100 years later and made them quite angry). However, this is still a pretty clear refusal to back down from the usurpation and a point of friction between the two empires.
So, overall, the Byzantines did care about the titulature in a non-trivial way, but pretty much always had something a lot more important going on, so never really did much about it.
I don't know much about the history of the term 'Byzantium,' which I have no problem using, so I won't gossip about half-remembered rumors of bad 19th-century British historians.
Sources: I'm basically just syncretizing some decent (though not without issues) general histories; I don't have a great bibliography for Carolingian-Byzantine relations specifically.
Gugliemo Cavallo (editor), The Byzantines
Mark Whittow, Making of Byzantium 600-1025 (UK title of the book is Making of Orthodox Byzantium)
Romilly Jenkins, Byzantium: The Imperial Centuries
Letter of Isaac Angelos: from Byzantium and the Crusades By Jonathan Harris (Hopefuly that links to pp. 133-4)
Followup question: the coinage of the term 'Byzantine Empire' comes from a German scholar in the 1500s, after Constantinople fell. Was this a deliberately political move to try to legitimize the 'Roman' part of the Holy Roman Empire (i.e., by calling the Eastern Roman Empire 'Byzantine', it meant that the HRE was the continuation of the original Roman empire)?