We currently make the distinction between the Republic and the Empire based on when several titles (Imperator, Dictator, Augustus, Caesar) were consolidated into one office, at least partially. How did Roman citizens of those times change the way they refer to the political entity of Rome? What are the different ways they referred to their "Empire" over time?
Thanks in advance.
Imperium Romanum (Roman Empire), Senatus populusque Romanus (the Senate and the People of Rome), Res publica (more commonly in the Republican era, but also used in the Imperial era). Not sure about all foreigners (I suspect some of them, in particular the ones being conquered, probably called the Romans very dirty things in whatever language they spoke), but Greeks would have used Basileíā tôn Rhōmaíōn (Kingdom of the Romans) or some variant thereof continuing on into the Byzantine era (when Greek became the dominant language in the surviving Eastern Empire).
Source: Four years as an undergraduate classics major, backed up by a quick spot check over on Wikipedia. Michael Grant is a good popular source on Rome, as I recall.
Chinese sources call Rome "Daqin", meaning Great Qin (like the Chinese dynasty). Gan Ying, an envoy sent west in 97CE describes Rome (without having been there) favorably listing all sorts of trade commodities and describing the system of government. He describes the people as "tall and honest" ... like "the people of the Middle Kingdom".