Bold parts:
a) True to some extent, but the people accepted that they were Roman citizens and identified as Romans after a few centuries. Add to the persection of everyone who identified himself as Greek (which = polytheistic heathen at the time) and the collapse of the west, everyone in the Empire identified themselves as Roman citizens. The term Roman would either mean Roman citizen or person from the city of Rome, not central Italy. It later on meant citizen of the city of 'Nova Roma' aka Constantinople. The interchangeable term was also Byzantine for the people from Constantinople as they stuck with the original name of the city. It's the only situation where the name Byzantine appears.
Except that they didn't.
Citizen has no meaning outside a republican basis. The term "citizen" stopped being used in the late empire. You are a "subject" of the emperor. The term "citizen" is not used in the Justinian Corpus (save to refer to local urban residents in the context of purely municipal affairs; urban residents of Rome are, of course, referred to as "Romans", residents of Constantinople as "Byzantines".)
Throughout the Corpus, it is always just the Emperor and his "subjects". The subjects are
not identified as "Romans", but rather merely as "subjects of the Emperor", or "inhabitants of Roman provinces" or "people under Roman rule", but never "Romans".
The term "Romans" is used in the Justinian corpus in only three ways:
(1) In the ethnic sense, to refer to the Latin-speaking residents of Rome & central Italy, the "the language of the Romans", etc.
(2) In the past tense, to refer to the ancestral peoples, kept at an arm's distance, i.e. "the Romans did this", the "Romans did that", the "Romans lost Africa, we conquered it", "the Romans had x, we should have that too", "the Romans founded the empire which we inherited"
(3) In the possessive form to refer to inherited institutions built by Romans but that remain contemporaneously ("Roman law", "Roman church", etc.)
The closest thing found in the Corpus referring to Roman citizenship is a small ambiguous passage on manumission in the Novellae; but other than that, usage is pretty consistent: "Romans" refers only to the inhabitants of Rome or to historical Romans, "citizens" are urban residents, otherwise everyone is simply "subjects".
b) That is untrue and I've never came across the ERE calling themselves anything else other than Romans, be it before 1080 or after. Care to show some evidence for that?
I am not talking about the ERE. I am talking about the
people of the ERE. The subjects.
1080 is the first official document we have referring the eastern people as "Romans", and the region as "Romania" (the "land of Romans"). That nomenclature only begins pushed with Alexius Comnenus. They did refer to themselves as "Roman Emperors" since the 7th C., but that says nothing about what the subjects were called.
"Graecia" is a well-defined region since antiquity. It is has been known as "Graecia" forever, and its people as "Greeks". There is nothing derogatory about the term. It is a geographical reality.
The region covered by the Byzantine empire through most of its history since the 7th C. covers the Hellenic world rather neatly, and nothing really beyond it. It is a natural appellation.
For the west, "Romans" and "Romania" was in central Italy. For obvious reasons, "Romans" meant that, and not the people of the east (except for a couple of isolated documents, e.g. one Anglo-Saxon doc from the 7th C.).
This whole "We are Romans" nomenclature was a late revivalist push. It was not how it was commonly known and, of course, resisted in the West because it caused confusion.
Interestingly, none of the Italian maritime states, who had extensive trade relationships with Constantinople and loads of treaties, used the term in their docs. It first shows up in the late 12th C. in Venice, and is only begins to be used with some regularity after 1204 to refer to the Latin Empire (although, curiously, Pisans & Genoese begin using it at the same time to refer to the exiled Greek despotates - I suppose to counter the Venetian-dominated C-polis). Otherwise, before this, references are almost always to "Imperator Constantinopolum", "Constantinopolitanum imperium" and "Terra Graecorum", "Graecorum imperator", or at best "Romanum Imperator Constantinopolum". In a couple of instances, even weird-ass "Imperator romeum" (a direct sound transliteration of the Greek word, rather than translating it to Latin).
Outside of the maritime states, in almost all the correspondence & accounts with the east with the Pope, the Frankish Emperors, the Anglo-Saxons, etc. it is never referred to as "Roman" until c. 1200. Pope Nicholas I uses it once only to admonish the Byz Emperor to stop calling himself that, Pope Leo IX used it once in "New Rome" form ("imperatori novae Romae") during an attempt to heal the Photian schism. All other western popes, kings, dukes & co., in all their massive correspondence, never used it at all. Nor in any other literature but a couple of direct translations of Greek texts.
The Norman & southern Lombard chronicles documents make use of the term occasionally after the 1080s (after it had become common use among Byzzies themselves), and then for Byz Italy alone (east is still Terra Graeca). But even so a rarity until the 1200s.
In the Crusader chronicles, "Romania" is used liberally - but most frequently to refer to Asia Minor alone i.e. explicitly excluding Constantinople & Greece! Crusaders were taking their terminology from how the Arabs/Turks used it ("Rum" for the Anatolian territory they roamed over - which is what the term was narrowed down to in Arab usage during the 9th C., BTW), rather than how the Byzzies wanted to use it (for the empire as a whole) Only in a couple of instances does it refer to the Byz empire (The weirdest case is a Crusader-era chronicle relating a 1190s address by Barbarossa, from "Imperatorum romanum Fridericum" to "Imperatorum romaniae Ysaac".)
The chansons de geste of the era, when it uses Roman in Byz refs, it is restricting it to those two narrow geographic senses: the Byz lands in Italy alone or Asia Minor alone, but never Constantinople, Greece or the Byz Empire per se. That is always "Greek".
So apparently the only people who called the Byzzies "Romans" were the Byzzies themselves late in the day and, for a short while, the Arabs. Whoop-dee-doo.
As for the church, since Christianity was a "universal" religion and not an ethnic one, like the ones everyone was used to at that point, the label was given according to language. And it was given by the westerners. The people of the ERE just called it the church, no linguistic or ethnic characterisation. For the ERE citizens, labelling something as Greek would instantly mean non-Christian. To call the Christian Church Greek would make no sense. It was a name given by the west which was adopted by the people around Constantinople way after the fall of the city to the Ottomans.
Everyone everywhere referred to, and continues to refer to, their church simply as "the church". It is only when you wish to clarify that you use distinctive adjectives.
Byzantine ecclesiastical documents are pretty consistent using "Greek Church" to refer to the east, and "Roman Church" to refer to the west, whenever they need to refer to them. There's no confusion there.
There's no secret plot. It is convenience.
There's no brainwashing going on, other than some self-deprecatory illusion you have that "Greek" is some sort of insult.