I recently finished Ancient Rome by Thomas Martin, and in the book he claimed that even after the rule of Augustus and the foundation of the Empire, the citizens of Rome continued to call it and consider it a continuation of the Republic, and that the separation between the two is more a modern invention.
But now I'm wondering, if this is true and the citizens continued to call their state a Republic, just when did they abandon the concept, or recognize that their form of government had irrevocably changed?
There is the word "res publica" meaning "state" or "public affair". Rome was a "res publica" since the Romans abolished monarchy, so a res publica is every state in which laws and traditions are respected by the rulers - the res publica is not a tyranny, but the "res populi", the "popular affair".
The Romans lived in a "res publica" for more than 400 years. But after the agitation of the Gracchi and the reforms of Marius, violence and butchery became common (three civil wars in 50 years) - the laws fell into desuetude, the power of the Senate and the people wasn't respected any more, as was the Cursus honorum, the rites of Roman religion or Roman law in general.
This is why Augustus, when coming to power by ending the last civil war, declared the restoration of the res publica ("restitutio rei publicae"). It's a matter of fact that the first Emperor did nearly everything to respect the rules and formalities of Roman constitution. Temples were renovated, the senate was purged of "indign" senators, the laws made under Augustus were confirmed by the popular assemblies that received a new voting building in Rome.
But even the most ignorant of the senators knew that the state Augustus created wasn't tha simple restoration of the old "res publica". They saw that the senate was purged of the opposition against the Emperor, that the Consuls held no power any more and that the popular assemblies had as much ponderosity as the elections in North Korea 2000 years later. That is why writers like Tacitus distinguished between the "res publica libera" (= republic) and the "res publica restituta" (= principate). Cassius Dio, wring in Greek, used "democracy" and "monarchy".
So, yes, as it was wrote in before, the Romans would have said, even after the fall of the republic, that they live in a "res publica" as we say to live in a "rule of law" or a "constitutional state". Though, Romans with some historical consciousness knew that the res publica of the 4th century AD was another res publica than the res publica of the 1st century BC.
As for the "Byzantines", I doubt they could identify with the concept of "res publica." They might have used "politeia", but "basileia" is much more probable. Greeks, more than Romans, were accustomed to live in monarchies since hellenistic times, and I mentioned the example of Cassius Dio, who, already in the 3rd century AD, spoke, referring to Rome, of a monarchy.