I'm sorry, but you are mistaken. The term "Imperium Romanorum" does not mean "The Roman Realm ruled by a Monarch (Emperor)" but simply "Roman Rule", that is to say, "the Realm of the Romans" or "the Government/State of the Romans". It was in use long before Augustus. The word "imperator" (meaning "leader, master, ruler") for someone who ruled the Roman Empire alone was first awarded to Augustus and became one of the many titles of the Roman Caesars, and much later, it derived the meaning of "emperor" in various language such as the French ("empereur"), the English ("emperor") and so on. But it did *not* have inevitable royal connotations to the Romans, at least not until rather later than the time of Augustus. When the Romans spoke of the Emperor, be it Augustus or Nero or Septimius Severus or Constantine the Great, they generally called him "Caesar". "Render unto Caesar that which belongs unto Caesar", and so on. In fact, the name Caesar evolved into a title too, and was the origin of the German "Kaiser" and the Russian "Tsar".
The fact that "Imperium Romanorum" could mean both "the Rule of the Romans" and "the Rule over the Romans", "the Roman government", was useful to the Caesars and led to the modern meaning of the word "empire".
It might be added that though Wikipedia is wrong in calling Julius Caesar "the first Roman Emperor", the Roman historian Suetonius (2nd Century A. D.) wrote a book called "Vitae Caesarum" ("The Lives of the Caesars"), which deals with Caesar and the first eleven Emperors. (The English translation published by Penguin is called "The Twelve Caesars".) Augustus claimed to be "the First Senator" ("princeps senatus") of the Roman Empire as a successor to Julius Caesar, who had adopted him and given him his name; being the First Senator also meant that you were the First Citizen. (And the word "princeps" also acquired royal connotations; it is the origin of the modern words "prince/prins/Printz/principe" et cetera.) Since all Emperors until the very end officially claimed to hold their position as inheritors of Julius Caesar, they all took care to assume Caesar as one of their names, if they did not already wear it because they were the son of the previous Caesar or adopted by him. For example, Tiberius, the second Emperor (or Caesar, to use the term employed by his subjects about him and all his successors), was officially called Tiberius Julius Caesar Augustus; before he was adopted by his predecessor and stepfather Augustus, his name was Tiberius Claudius Nero.
It would make as much sense to say that the Roman word "imperator" meant "emperor" as to say that the modern word "leader" means "king". In fact, Augustus and his successors presided over an elaborate charade to the effect that they were not royalty but simply ruled Rome as successors to the great Julius Caesar. Of course, that was not true, but to formally declare oneself King (Latin "rex") would have meant courting suicide. Julius Caesar was murdered because he was suspected of planning to proclaim himself king. Inevitably, pretty soon everyone knew that they *did* have a monarchic ruler, but it was wisest to pretend not to know it...
Literally, "res publica" meant "public matter, public thing", that is to say, "the state". The word said nothing about the nature of government. The modern meaning of "republic/Republik/repubblica" and so on is very late and derived from the French "république". That the meaning "polity without a monarch" is very late can be seen, for example, from the fact that Poland, which was ruled by Kings, was officially called "the Polish Republic". The Latin word for "democracy" or "republic", that is to say, a state without a monarch, was simply "populus" ("people"). Populus Romanus = the Roman People (State). But Augustus claimed that he simply led the "Populus Romanus". He wasn't a king, all he had done was to unwillingly, in view of the evil and chaotic times, assume a lot of power as the First Citizen, so the populus was still supreme. He just happened to preside over them. See?