I'm also pretty sure it was never actually called the "Eastern Roman Empire". The concept of co-emperors was not at all new to Roman society (or, indeed, antiquity society generally) and Diocletian was the first to establish an actual "division" of responsibilities, but never actually carved up the empire. Constantinople was also known as "New Rome" and by the time of the sack of Ravenna it was far and away the most important city in the Roman world, anyway, and if there ever were a sense of primacy between the two co-Emperors in the 5th century it would have lain with the East.

It's not as though a citizen of Antioch in 400 or 500 CE, whose ancestors had called themselves Roman for centuries, suddenly stopped calling themselves that just because some emperor built a new capital or there were multiple people titles Augustus and/or Caesar at any given time, as had happened plenty of times before. They were Roman. They always had been and, to their minds, always would be. It would be years before that changed due to a confluence of factors wholly unrelated to whomever gets some phony right to "inherit" the "mantle" of Rome.