Well, what we call Imperium swallowed what we call the Republic. For them the two concepts were broader enough for that to make no sense.
I mean, seriously, for them "imperium" was just one out of 4 or so kinds of power that they had different terms for. (They were pretty picky and specific about whether your power was auctoritas, or potestas, or imperium, unlike today when we make a hash of it.) It was used all the way from kingdom, through republic, and all through the empire era.
Unlike later, when everyone scrambled to define empire in a way that lets them claim to be like the Romans, the Romans made no pretense of it being based on anything else than that they had the state power to command in that territory. And even that use of "imperium" to mean "the territory over which we have imperium", was really a figurative use of "imperium".
If you want a term that became more associated with imperial power, try Imperator, which was appropriated by the emperor as early as Augustus. (Previously, anyone who had imperium was an imperator.)
And the "res publica" can't become anything else either or disappear, because it meant "public matters" or "public affairs." It didn't mean it had to be run by the public. It was just the "res", as in a concrete thing not something abstract (see how it got us "reification"), and "publica" as in belonging to the public sphere. Compare with "res privata" (private matters, or "property") or "res militaris" (military matters.) It's not about who runs it, it's about what
domain of "res" you're talking about.
It wasn't even used to mean republic until well into empire times, when it was
sometimes used
figuratively to mean how things used to run back then.
Basically, what I was trying to say with that message to gumboot was more like let's not make a hash of our concepts and the Roman concepts. Yes, we can say that the Empire replaced the Republic, by our concepts, and whether that fit the words and concepts the Romans had for it, well, who cares?
