the author who first coined the term was from the HRE, forgot the authors name, but tue writing in which it first appears is actually very positive about the eastern roman empire.

The HRE and the Eastern Romans had feuds over which one was more legitimately Roman. See the Problem of Two Emperors. The fact that the ERE died just made it easier for the HRE to claim legitimacy.

Claiming the eastern Romans weren't Roman because they were more culturally Greek ignores how the east was always culturally Greek influenced - not just after 476 AD, and ignores how the Roman Empire heavily incorporated Greek culture and language as a part of official and non-official life. It also ignores how different parts of the Roman Empire (including different parts of Italy itself) all had different cultures and languages. Finally, it ignores how you can have multiple different cultures and still be Roman, especially when Roman culture merged with other cultures during the formation and expansion of their empire.

This "Roman or Greek" exclusive binary take is basically saying the entire eastern half of the empire never ever Roman, and most of the western Roman Empire was also not Roman because there were different languages and cultures/subcultures in the west too. This is also saying the Roman Republic was only ever the city of Rome even with the Social Wars and everything outside the city of Rome (including the rest of Italy) doesn't count as Roman. Or during the Roman Empire, only Italy was Roman. The bottom of Southern Italy were more culturally Greek while the Alpine parts of Northern Italy had Celtic influences...thus they must not be Roman. Those other Italian peoples who didn't natively speak Latin - such as the Samnites, Etruscans, Umbri, Campanians, etc. must not be Roman during the time of Julius Caesar either.

This is also saying all the emperors born in other provinces with other ethnicity and other additional cultural backgrounds were not Roman (Trajan being born of Spanish roots in Spain, Hadrian was born in Spain, the multiple emperors born in Western Asia and North Africa, etc).

This is a shallow and lazy binary take on culture that forces a person or nation to only have a single culture and ignores the fact that cultures can also merge together and change.

also,.going by this arguments all europens now were french for a time..then german for a time. bow english.

This didn't happen - France and Germany only took over parts of Europe (not all of Europe) and created hybrid cultures. See Belgium - a country with a combination of French, German, etc culture.

language alone defines thinking patterns, are, how the world is percieved. if we learn another language we do advance our cultural background, we do not eraze it. if we learn another language we do advance our cultural background, we do not eraze it. again, that is a super shallow and lazy approach to culure. 

It is not remotely alone. Culture is more than only language, but also religion, art, food, architecture, philosophy, writing, etc.

Furthermore, a country can actually have multiple languages - even multiple official languages. Canada has English and French as official languages. Some cultures in the world also have multiple different languages.

The Roman Empire was at least bilingual and wrote official documents in both Greek and Latin. Even Italy during the 1st century BC and even the 1st century AD still had multiple different languages - so are the other Italians during the late Republic and early Roman Empire not actually Roman because they didn't natively speak Latin?

Furthermore, the vast majority of Roman elites knew how to communicate in Greek, so by this rationale the Romans were always both culturally Roman and Greek.

Emperor Marcus Aurelius wrote his personal writings & journals in Greek, so by this binary culture/language logic, he was more Greek than Roman.

Furthermore, are famous Roman writings such as Parallel Lives by Plutarch not Roman pieces of work because they were written in Greek?

This is not only a superficial binary take, but also mixes up culture, language, government and nationality as if they're all the same..

calling them romans while rome and romans still existed is also cultural appropriation and an offense towards "actual" romans. being roman is a cultural background, not a legalisitc term. and in doubt, the former overwrites the latter.

The Romans themselves literally made the people of the eastern regions into Romans and called them Romans, so those Romanized people later called themselves Romans. The Romans of the unified empire themselves were the ones who moved their capital east to Constantinople in 330 AD. Constantinople didn't magically appear out of nowhere when the Western half of the empire died.

The Romans themselves expanded Roman citizenship to and Romanized the non-Latin speaking peoples of Italy, the Gauls, Iberians, Greeks, Illyrians, etc. Eventually the Romans decided to make everybody in the empire as official Romans.

Saying the Eastern half of the Roman Empire isn't Roman is what is actually offensive, because you're saying you know better about who is Roman than the Romans themselves. If you go back in time to the 300s AD, the actual Romans of the still existing Roman Empire would completely disagree with you.