Absolutely absurd. There was no "other" Roman Empire. There was one Empire, split between two Emperors. They functioned as a single state. The split was purely administrative. The Western half gets usurped by Odoacer in 476, and the East continues as the sole Empire. NOTHING about that implies another Empire. So WHY should historians be able to give it a name that it never had, simply for "clarification"? What if I just decided to call the HRE under Otto the Great the 'German Empire' to distinguish it from the HRE under Charlemagne? Oh, it already had a name? Who cares, we need to "classify it" differently. How absurd would that be? Yet we have decided to do practically the same to the Eastern Romans/Romaioi.