The term 'Byzantine' refers to the small town of Byzantium that became the capital city of the Eastern Roman Empire, Constantinople. The town itself was small but not minor, and a good deal of historical records exist referencing the place before Constantine the Great rededicated it in 330 and the city was renamed 'Constantinople'. Inhabitants of the city might have referred to themselves as Byzantine before that point, as city or town loyalty was as important as national allegiance in the ancient world (and medieval world for that matter).

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What I think you're getting at though is, when the Western Empire fell, did the Eastern Empire see themselves as Roman? And the answer is simply: yes. In fact, the 'fall' of the Empire was seen as no such thing: even in 478 when Odoacer returned the crown of the Western Emperor to Constantinople, declaring there no need for a new Augustus of the West, he also made clear that he served the Eastern Emperor in doing so. When deposed by Theodoric the Great later on, the Ostrogoth ruled the Kingdom of Italy on behalf of the Emperor. For most Romans, this was a return to 'One Emperor Rule' which dominated most of the Roman Empire's history. Not much had changed: they were Romans living in a Roman Empire, albeit in a kingdom managed by a foreigner (though even then, Romans were used to this. Half a dozen great foreigners ruled Roman territories on behalf of Romans by the fifth century).

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The first time Romans might have seen a decisive break in their history is when Justinian the Great declared a reconquest of Italy. If Italy was to be reconquered, then that must mean Italy was lost at some point. Inhabitants must have been surprised to say the least: you mean we were not Romans? Ultimately, modern anachronistic nationalist tendencies makes comprehension hard, but Romans wouldn't have thought of themselves as not Roman even if they were not residents or citizens of the Eastern Roman Empire. Even if you were a Roman living in Italy, perhaps under Totila in the sixth century, you would have thought of yourself as Roman.

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Up until the fall of Byzantium in 1453, inhabitants thought of themselves as Romans. Study of the Byzantine Empire is recognizably and recognized as woefully lacking. The wealth of the Byzantine Empire was well above any other kingdom in Europe. We do know the Empire throughout its life thought of itself as the true Roman Empire, however, and its citizens saw themselves as part of that history.

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The question arises, then, where does the term 'Byzantine Empire' come from? It arose a century after the Empire's fall in the work of Hieronymus Wolf, popularised in later European work. This division, strictly speaking, is solely a European one: the Chinese and Arabic-speaking world tended not to make a distinction between the Roman and Byzantine Empire, though modern scholars tend to do so more often simply out of ease of reference and so there's one global division. More modern work talks about the transition between Rome and Byzantine Empires as a smooth one and criticises the implied argument that the two Empires are clearly distinct in any way. In short, however, the Byzantine Empire is a product of modern historians who want to more clearly distinguish between two periods in Rome's history, much like the division between Roman Republic and Roman Empire.

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EDIT: I forgot to mention some good sources for this! While this answer is pretty much a consensus that you can get in all good books on the subject of the late Roman Empire or early Byzantines, I am currently reading John Julius Norwich's Byzantium: The Early Centuries, and it's a wonderful popular history book for this time period. Easy to read, accurate, engaging, and chock full of interesting political history of the early Empire.

What you have heard is correct.

Specifically, the term "Byzantine Empire" comes from the German historian Hieronymus Wolf. In 1557 he published the Corpus Historiæ Byzantinæ, which was mostly a collection of Byzantine sources. It is not completely clear why he decided to call it that, but it may have been that he refused to accept that the Byzantine Empire as the true continuation of the Roman Empire because that was a claim the Holy Roman Empire also made, and Wolf was part of the HRE. It was a convenient bit of revisionism. Because Wolf had basically written the sourcebook, many early modern historians consulted the Corpus Historiæ Byzantinæ for source material and picked up the term themselves.

The term stayed in France and Germany until Edward Gibbon. As with quite a lot of Roman historiography, Edward Gibbon played a major role in defining how modern historians think about the Roman Empire. In his Decline and Fall of the Roman Empire, he does not really accept that the Eastern Roman Empire was truly Roman:

"Whatsoever changes had been introduced by the lapse of ages, they alleged a lineal and unbroken succession from Augustus and Constantine; and, in the lowest period of degeneracy and decay, the names of Romans adhered to the last fragments of the empire of Constantinople."

To him, the term Byzantine worked brilliantly to distinguish the greatness of Rome under Augustus and Trajan from the pathetic rump state that emerged following the Arab Conquest. In his writings on The Crusades, he called them the Byzantine empire, and the word caught on in western historiography.

Generally speaking, we still use the term Byzantine because it's a valid distinction to make. Politically, socially, culturally, linguistically, and militarily, the empire left behind by the emperor Heraclius in 641 was fundamentally different from the empire he took control of in 610. He changed the official language of the empire to Greek, he reorganised the empire into Greek Themes rather than the old Roman provinces or later Roman diocese (As /u/Anthemius_Augustus has pointed out, these are two misconceptions that for some reason continue to crop up even in modern scholarship, including my sources. Heraclius did not mandate a change of language to Greek and evidence that he was behind the Theme system is shaky at best). Heraclius did institute major military reforms, a restructuring of the economy, reforms to the administration of the empire etc. during the Last Great War of Antiquity to cope with the existential threat of the Sassanid invasion. This, coupled with the severe loss of territory during the Arab Conquest, constitutes enough of a drastic shift in the fundamental dynamics of the empire to say that it became something different enough to deserve a new label despite its legal continuity. Obviously, there were also many more gradual changes, but Heraclius and the Arab Conquest tend to be the usual cut-off point because economically, militarily, and demographically, it was substantially different to what came before. To a lesser extent, this distinction gets made for the later Roman Empire as well - historians generally feel that Rome before the Third Century Crisis was different enough from the empire that emerged from the end of it to have the word "later" put before "Roman Empire" from the 280s onwards.

So what did the Byzantines call themselves? Romaioi - Romans. The elite were very touchy about this; in 968 envoys from the Pope were imprisoned for calling the emperor "Imperator Graecorum" rather than "Imperator Romanorum", and it derailed negotiations between the Holy Roman Empire and the Byzantines:

"to increase my calamities, on the day of the Assumption of the Virgin Mary the holy mother of God (August 15), there came-an evil augury for me-envoys of the apostolic and universal pope John, through whom he asked Nicephorus ,the emperor of the Greeks " to close an alliance and firm friendship with his beloved and spiritual son Otto "august emperor of the Romans." Before the question as to why- this word, this manner of address, sinful and bold in the eyes of the Greeks, did not cost its bearer his life-why he was not annihilated before it was read, I, who, in other respects, have often shown myself enough of a preacher and with words enough at my command, seem dumb as a fish! The Greeks inveighed against the sea, cursed the waves, and wondered exceedingly how they had been able to transport such an iniquity end why the yawning deep had not swallowed up the ship. " Was it not unpardonable," they said, "to have called the universal emperor of the Romans, the august, great, only Nicephorus: "of the Greeks"';-a barbarian, a pauper: of the Romans'?"

(The full 10th century account of this whole disaster can be found here: https://sourcebooks.fordham.edu/source/liudprand1.asp)

Over time this sensitivity softened; the emperor Alexios I Komnenos did not mind being called Imperator Graecorum and his daughter Anna Komnena referred to herself as a Hellene as well as a Roman. She had a particular love of the Greek classics, and identified herself in accordance with that, though she still called the empire Roman and her father the emperor of the Romans. They also sometimes referred to the empire as oikoumene - the civilised world. This was based on the idea that the Roman Empire was civilised and the rest of the world was barbarian. Anna Komnena referred to the crusaders as Franks and Kelts - as if they were barbarians.

So "Byzantine Empire" is indeed a more recent invention. They called themselves Romans.

I’m not a common poster on this sub, but I’ll try as hard as I can to give a good answer here.

First off, you would be correct that “Byzantine” is a term invented after the actual empire to discuss it without confusion. In medieval times, there was no shortage of entities with a partial or legitimate claim to the title of “Roman”. The major entities of this title would be the Roman Empire (or Byzantine Empire, as it would later be called), the Roman Catholic Church, and the Holy Roman Empire. Western scholars had a hard time distinguishing between these in their time, so they would usually refer to the Roman (Byzantine) Empire as the Greek Empire, due to its location.

Internally, though, is where things get even more confusing. The majority of Byzantine citizens considered themselves to be Roman, and that was the demonym used. However, as the Western Roman religion and the Latin language began to die out in the empire, many citizens referred to themselves as “Hellenes”, or Greeks, to better represent their identity. The Byzantine emperors began to fashion themselves as the “Emperor and Autocrat of the Romans”, furthering the link to the Roman identity. At this point, new terms began to be used both in the east and west of Europe to refer to the empire colloquially, namely “Romania” and the “Eastern Roman Empire”.

When speaking in a historical context, feel free to say “Byzantine” to make things clear, but the terms used in the era of the empire itself would likely have been “Roman” or “Greek”, or later “Romanian” or “Eastern Roman”. There is truly no good label to use without its faults, so just try to be as clear as possible.