Hi Lele,
I disagree with you. Just listening to a few podcasts by Barbero is enough to understand why you shouldn't feel offended at all. The Eurocentric vision that took hold in the early and late Middle Ages was unknown to the Roman Empire. The elite that dominated our peninsula after the invasions became barbarian, and over the centuries, even those who once considered themselves Roman started calling themselves by other names. In the eyes of the original elite of the Empire, which had not been "contaminated," the West was lost, a place where the barbarians ruled. It was the Pope, for his own interest in legitimization, who orchestrated the charade of the Carolingian Empire. Since Constantine, the power of the emperor among men was seen as coinciding with that of God. In the East, it was like this. The emperor was the highest authority, but the church was autocephalous, as per tradition, without a central authority. The Pope, however, insisted on the primacy of the Church of Rome, and over time came to view his authority as equal to, and later superior to, that of the emperor!
In conclusion, since the topic is about how to name that Eastern Empire in a game, and considering there are 74 pages (as of today) of discussion, I'm sure they'll just make a button to switch depending on everyone's preferences. After all, we're talking about this on a computer game forum, not at an academic seminar.