It's even more complicated than that. The Byzantine Empire was conquered by a group of crusaders in the Fourth Crusade in 1204, with several rump states cropping up. The Empire of Nicaea reconquered Constantinople in 1261, so people usually call it the continuation of the Byzantine Empire. By the 1400s, this Byzantine Empire quite literally existed at the whim of the Ottomans, even if they only entered Constantinople itself in 1453. The third rump state of the Byzantine Empire was the Empire of Trebizond, whose last bastion in Theodoro held out against the Ottomans until 1479.

And, by the way, practically none of these countries were so named to their contemporaries. Basically everyone I've mentioned here considered themselves the Roman Empire.

The Byzantines not call themselves Byzantines; they considered themselves Roman. See a helpful answer on Reddit¹ which notes an episode in which the Byzantines imprisoned an envoy for calling their emperor that of the Greeks. It would of course be obtuse to call the Byzantine Empire the Roman Empire, but the renaming was after 1453.

¹ https://www.reddit.com/r/AskHistorians/comments/ak435w/did_t...

This reminds me a lot of bigger companies (IE Microsoft and their "lost decade") that lose time and experience and product margin to their competitors but that get back after a while- the Romans had so much consolidated power that, if it weren't for people jockeying for power, they could've survived in a shrunken, but still pretty huge form.

> The Eastern half got renamed to the Byzantine Empire

That's not correct, Byzantine Empire is the modern name we give to that empire during the Middle Ages. They called themselves just Roman Empire, in their minds they were just the same original Roman Empire but headquartered in Constantinople.

To be clear, we're the ones that renamed it the Byzantine Empire. As far as the people who lived there were concerned, they were living in the Roman Empire, and they would not have understood in the slightest why anyone would think otherwise. Constantine moved the capital, and there was no disruption in the government and administration of the empire until the Fourth Crusade, although they did lose a tremendous amount of territory in the early Muslim conquests.