As you alluded to, Constantinople was originally called Byzantium, when it was founded all the way back in the 7th century BC. About 1000 years later, emperor Constantine renamed it after himself in the 4th century, and it became the capital of the eastern half of the Roman Empire, and then of the entire Roman Empire when the western half ceased to exist.

The term “Byzantium” was reintroduced by a 16th-century German historian named Hieronymus Wolf, essentially just as a shorthand way of referring to the empire, which otherwise didn’t really have a name. Well, that’s not quite true, it did have a name - its own inhabitants still called it the Roman Empire, or just Rome. But western Europe had been denying the Byzantine Empire’s relationship to Rome for many centuries already, going all the way back to Charlemagne in the 9th century. Medieval western Europeans called it the “empire of the Greeks” or the “empire of Constantinople”, specifically to deny that that weird empire over the in the east had anything to do with Rome. Surely the heirs of Rome were in the west! That’s where Rome was, that’s where people spoke Latin (or its daughter languages), and that’s where the Latin-speaking Roman church was, etc…not like the decadent east where they spoke Greek and had a different kind of church. Those strange Greeks over there were so weak that they couldn’t even protect their empire, which had fallen in 1453, a century before Wolf was writing. Why would they deserve to be associated with the strong and honourable Romans?

According to the Byzantine historian Anthony Kaldellis, Wolf’s use of the term was actually pretty obscure until the 19th century. Before that, historians like Gibbon in the 18th century still considered the Byzantine Empire “Rome” - Gibbon’s “Decline and Fall” goes up to 1453 - but a Rome inhabited by Greeks, not Romans. Kaldellis has recently argued that “Byzantium” was really only popularized in the 19th century because of the political situation in Europe:

“…among the reasons for this switch to Byzantium was the creation of a modern Greek state in the 1820s. By midcentury, the Greeks were agitating to reclaim Constantinople and recreate the “Greek empire” that was their birthright. Because such a state would have come at the expense of the Ottoman empire, the weakest link in the fragile post-Napoleonic imperial order, western powers were opposed to this idea, and some even perceived it as a Russian plot. The Crimean War, which was fought against Russia by France and Britain, two states that played a foundational role in the development of western scholarship about Byzantium, led to an intensification of Russophobia in the west and the fear that the tsar was using his Orthodox clients in the Balkans to create a Russian-dominated Greek Orthodox empire centered on Constantinople. The historiographical term the empire of the Greeks could be seen as legitimating these aspirations and fueling the Greeks’ Grand Vision of imperial instauration. ‘Byzantium,’ by contrast, was more neutral and less inflammatory and gradually replaced the ethnic name for the empire after the Crimean War.”

Using “Byzantium” means that, 1) we can pretend they had nothing to do with Rome, and b) that they also had nothing to do with the modern Greeks. The independence of Greece from the Ottoman Empire, and what to do with Ottoman territory when the empire collapsed, was a politically sensitive topic in the 19th and 20th centuries. Giving academic weight to the idea of a “Greek empire” would have simply encouraged the modern Greeks to recreate their empire, so that term was avoided. And the term “Rome” was also avoided because, well, that was just a natural reflex for western Europeans - for almost a millennium already, the west had been systematically denying the Roman-ness of the Greeks in the east.

“Byzantine” was therefore

“A fictitious generic category that avoids the complications of both the Roman and the Greek labels.”

It sounds a bit like a kooky conspiracy theory…but it kind of is. It was a conspiracy that spanned centuries, and eventually won out because of the way that academic history/historiography developed in Europe and spread to the rest of the world.

Source:

Anthony Kaldellis, Romanland: Ethnicity and Empire in Byzantium (Harvard University Press, 2019), particularly the introduction, and even more specifically pages 15-16, where these quotes come from