One of the features of language is that many words or phrases have more than one meaning, and people determine which meaning is meant by context. In particular, the phrase "Roman Empire" can mean either Eastern and Western branches, or just the Western one. In this case it is obvious it is the latter that is intended.

However, it is good to bring up Byzantium since so many people are completely unaware that it was the Eastern Roman Empire.

New Latin is still being invented, in academic circles, for naming new things.