Couple of misconceptions here, but I'll go ahead and answer the actual question, first ;) Citizens of the early Roman Empire called the State "res publica," just as they had for centuries beforehand (for one example, see Seneca's 14th letter to Lucilius, dating to the mid first century CE). Now I'm about to say something that'll seem kinda dumb, but bear with me. It's always good to remember in these situations that the Romans did not speak English. While we refer to things as the Roman Empire and the Roman Republic, it's mostly a distinction by historians. The Romans themselves, if they were specifically talking about the State/government, just referred to it as res publica - "the public matter." Our English term "Republic" means something entirely different, in that the people elect representatives to the seat of power, in which they make laws etc.

The actual levers of power are kind of important here, though, and that's what you're certainly getting at. The people of Rome were well aware of Augustus' power and control over the levers of government, but it's easy to not get too riled up if you see all of the traditional bits of government working just fine - better, in fact, than they had been for the past century or so. Elections still happened, even if the person who was favoured by the princeps tended to win, and the princeps himself was often elected to the consulship. Outside the city of Rome itself, local elections were still run, dominated by the upper class. Pompeiian graffiti shows us the extent of these campaigns, with electoral propaganda all over the place, indicating that elections locally were still notable and eventworthy, and possibly even contested (you don't go all out to advertise a political candidate who has no opponent). Local governments were often even still run under their own auspices, indicated by one of Pliny's letters to Trajan:

Lex Pompeia, domine, qua Bithyni et Pontici utuntur, eos, qui in bulen a censoribus leguntur, dare pecuniam non iubet; sed ii, quos indulgentia tua quibusdam civitatibus super legitimum numerum adicere permisit, et singula milia denariorum et bina intulerunt. Anicius deinde Maximus proconsul eos etiam, qui a censoribus legerentur, dumtaxat in paucissimis civitatibus aliud aliis iussit inferre. Superest ergo, ut ipse dispicias, an in omnibus civitatibus certum aliquid omnes, qui deinde buleutae legentur, debeant pro introitu dare. Nam, quod in perpetuum mansurum est, a te constitui decet, cuius factis dictisque debetur aeternitas.^1

Bolding the Latin for importance's sake, with the English below, since the Latin is what we're actually looking at here. If you read the English, you'll note a marked deference to Trajan, present in all of Pliny's letters, especially with the word "dominus." This word generally translates into "master" or "lord," and begun to be used with reference to the Emperor during the so-called "Dominate" period, begun by Domitian. Here, we see the Emperor begin to be referred to as king in all but name. Now, the other things that I highlighted are super interesting, since Pliny here refers to the local council/senate/whatever as a bule - a transliteration of the Greek βούλη. It basically translates to "city council," which is essentially the same idea as a local senate (in fact, in Trajan's response, he specifically uses the word "senatus," rather than "bule"), but it indicates that the cities in Pliny's province (Bithynia and Pontus) were still governing themselves by the same systems that they had been since the time of the Successor States - and that's what most people outside of the city of Rome would refer to when talking about government. When referring to the Emperor himself, people in the East generally just cast him in similar terms as they had the Hellenistic and Persian kings - something that goes back to Pompey the Great being hailed as "King of Kings."^2 You occasionally see this with Roman Emperors as well, but most important is the constant association with divinity ("son of the god" or just "the divine"), an emphatic tradition spanning centuries.

tl;dr, depends on who was doing the referral and what level of government they're talking about. Also what year, since context changed gradually over time. Hope that answers your question!

1. Pliny, Letters, 10.112. [Translation lightly edited by me, since the Loeb version is meh]. Bithynia and Pontus, milord, are subject to the code of law drawn up by Pompey, which makes no provision for payment of entrance fees by those elected to the local council by the censors; but in certain cities, where persons in excess of the legal number have been nominated by your special permission, they have been paying fees of one or two thousand denarii. Subsequently the governor Anicius Maximus, made it a rule (though only in a very few places) that persons elected by the censors should also pay an entrance fee, which varied from city to city. It remains then for you to consider whether from now on all persons elected councilors should pay a fixed sum as entrance fee; for it is only fitting that a ruling which is to be permanent should come from you, whose deeds and words should live for ever.

2. For more on this, see Warwick Ball's Rome in the East.