It's a good deal more accurate to say that what you just posted is a modern invention. From what I've read, the Byzantines were pretty ambivalent about their identity. The idea that they didn't use 'Byzantine' is a particularly outrageous fraud - to take just one example, have a read of Procopius' Secret History. Steady use of 'Byzantine' and 'Byzantium' to refer to the city and it's inhabitants - which we have good evidence was indeed extended to refer to the state itself. Random example, with Procpius describing, I believe, the behaviour of the Blues:
What else do you think they called themselves? Constantinopolitans? I really don't know where this idea that the Byzantines were a bunch of macho-weirdos who wanted to out-Rome the Romans comes from.
Leaving aside the question as to whether Procopius should be called a "Byzantine" as opposed to a Late Roman this is by and large untrue, Byzantium is used for the city, for the Empire "Roman" is more common. And Mark Whittow finds an interesting example of a shipwrecked Byzantine who does not even say that, but simply says he is a Christian
Certainly the sources are translated as "Roman" not "Byzantine," and by sources I mean the Alexiad, Psellus, Choniates, Three Military Treatises, Book of the Eparch, Luke the Stylite, De Administrando Imperio, Famers Law and god know's what else I have long forgotten. And unless there is some bizarre conspiracy going on I suspect that is because they say Roman not Byzantine. Considering how little we have to go on and how every possibly inflection and interpretation in the original material is analysed to death this would seem the most bizarre thing for the historians.
Edit: Also, use of "Byzantine" in the sources need not indicate everyday usage - the Byzantines loved archaising, they considered the cliche the highest form of art (someone should tell Martin Amis about that). The Battle of Crecy was referred to as a fight between the "Britons" and the "Celts" for example.