I've been reading Thomas R Martin's Ancient Rome, in which he claims that the designation of republic and empire is an anachronism, since Romans, throughout their history, never stopped referring to Rome as a Republic.
However, it seems like Romans would likely still have noticed a pretty drastic shift in power structure, and there were movements to restore the system of the republic (Caesar's assassination, I would assume more in the post empire period), which would Imply they perceived the republic and empire as two distinct time periods.
To add some more specific questions:
Did they perceive the republic and empire as two distinct time periods?
If they did, would they have put the division earlier (due to the chaos of the late republic) or later (due to Augustus's principate)?
Was there any support for re-establishing the republican system later on in the empire (under the five good emperors for example)? This was portrayed in the movie Gladiator.
Under Augustus's principate, to what extent would the average Roman have been aware that the republic was effectively over?
I know that's a lot of questions to throw out there at once, but i find the period quite fascinating and would be very grateful for any answers at all.
I think the argument based on vocabulary is a little absurd. Res publica is how the Romans refer to their government/politics but it literally means "public affairs" and clearly did not denote one single type of system like "republic" does today.
Everybody, and I mean everybody, recognized a shift in the way Roman government ran between Caesar and Augustus. The idea that there were people in the city of Rome who figured the res publica was the same under Augustus as it was under Pompey is, quite frankly, insulting to Romans. Augustus slowly gained public support for over a decade before going to war with the other triumvirs. Once he one, the political elite, fatigued by decades of civil war went along with him because it was better and cheaper to live in peace and not have to spend money trying to get elected.
Ancient historians do see Augustus as a transformative figure. In fact, Augustus claimed to be a transformative figure himself. He physically transformed the city of Rome in part by building up the largest public spaces in the city. Tacitus's Annals was written in part to show how terrible Roman values had become after memory of the republic had faded.
Modern historians date the founding of the Empire to 27 BCE when Octavian took the name Augustus. This is picked somewhat arbitrarily but it shows his total control. Caesar was still operating under Republican precedent (he hadn't done anything Sulla didn't do before him) and after he died Rome was ruled by the "Second" Triumvirate which divided the empire into three units and gave one person near-absolute authority over those units. This ended in 32 BCE after Octavian defeated Marc Antony at Actium. At that point Octavian was the sole ruler of Rome but in 27 BCE the Senate essentially made it official.