I think they should be separate for the game, simply as an outsider reference.
Take for example modern day China and Taiwan, technically they are both China, but they are legally two separate countries and the rest of the world refers to Taiwan as Taiwan rather than China, even though it is China.
In EU4 it should be possible for any nation with legitimate claim (former Roman state), to take Rome and form the Empire of old, and destroying or conquering Byzantium shouldn't be a necessary requirement of that. So for clarity, we need two names, one for the Eastern part of the Roman Empire and one for the Western part of the Roman Empire.
Though personally, I would like to see a different name for the Western Roman Empire as well, then a reunification mission/event to combine West & East into one again for the full Roman Empire tag.
Take for example modern day China and Taiwan, technically they are both China, but they are legally two separate countries and the rest of the world refers to Taiwan as Taiwan rather than China, even though it is China.
In EU4 it should be possible for any nation with legitimate claim (former Roman state), to take Rome and form the Empire of old, and destroying or conquering Byzantium shouldn't be a necessary requirement of that. So for clarity, we need two names, one for the Eastern part of the Roman Empire and one for the Western part of the Roman Empire.
Though personally, I would like to see a different name for the Western Roman Empire as well, then a reunification mission/event to combine West & East into one again for the full Roman Empire tag.