One reads frequently that the Greek-speaking people of the Empire whose Capital city was "New Rome which is Called the City of Constantine" [the official name of the city until 1922 I think?], never used the word "Byzantine".
This is quite untrue - many Byzantine historians (Anna Comnena and Nicetas Choniates for instance) use "Byzantine" frequently, as any scan of their texts will show. It is true, however, that they never used "Byzantium" and "Byzantine" in the synecdochal way that modern Westerners do, but untrue that the word would be unfamiliar to them. By this I mean they used the term to refer to the city and inhabitants of Constantinople, not the entire East Roman realm.
In the 16th and 17tn centuries "Byzantine" becamse a useful word to refer to the Medieval Roman Empire in the East. [This was especially the case with the work of the great scholar Du Cange, who popularised the use of "Byzantine" when he was in fact talking about families from the city of Constantinople, a usuage well-justified by Byzantine sources.] The later and more general use of the word has long proved useful to just about everyone, Greek and non-Greek, since the Greek-speaking agricultural empire of the 7th century and later, with more or less only one major city for centuries after, was clearly distinct from the multi-civic-centered, multi-linguistic Roman Empire of antiquity. This is not to deny important formal continuities.
With the use of "Roman", which, I have to say, reminds me of the use of the word
"Anglo" by and about some distinctly non-English populations in Californian usage, it is true that this was the *normal* term used by Byzantine historians writing about their political history. I am unsure about how widespread that term would be in common usage - since Byzantine historians were rather too fond of archaic terminology to described various peoples ["Medes" and "Scythians" turn up, for God's sake!]. I do recall venturing into a Church in Istanbul in 1983 and greeting the custodian with some phrasebook Turkish, only to hear the thrilling response "No No, I am a Roman".
Then we come to "Hellene". There is no doubt that in normal Byzantine usage it meant "pagan". Since the continuity of Greek populations in almost *any* part of "mainland" Greece is extremely hard to prove, I am very unhappy with a suggestion that 17th century Greeks were "still" using "Helene". [And this is not just a matter of the famous 8th century Slavonic population movements, but the much later effects of plague and piracy - post 1347 accounts of the Morea, for instance, make it clear that thousands of Albanians were "imported" to farm empty land.]
"Hellene" seems only to have assumed a positive sense only under the influence of proto-nationalism. There were clearly some groups in the last centuries of Byzantium - Gemistus Plethon is the most famous representative - who tried to outline a distinct "Hellenic" identity. The mutual interaction between Greeks in Italy, where I presume "Hellene" received a boost from certain Renaissance schools of thought, and Greeks in Greece might also have had an effect. If sources for a 17th century use is "intellectual", then it might very well represent this late Byzantine approval of "hellene" rather than any "continuing" ancient use.
Some have argued that it is misleading to ever refer to the Eastern Roman Empire as Byzantine, with the claim that it is and an anachronism.
But the problem this is that there is not a single example*I know of where any
contemporary sources calls "the empire whose capital city was New Rome",
the "Eastern Roman Empire"!
Let's be careful in what we are suggesting is anachronistic here. And when it comes to terminology, as I pointed out above, the Byzantines (or at least the Byzantine historians) are the last people in the world who can insist on "politically correct" words. [Personally, however, I do not object to being called a "Frank"!]