Caesar official turned the Roman Republic into an Empire, but even before him, Rome ruled over many different people much like an Empire. Could Rome be considered an Empire before Caesar?
What you are noticing is a confusion of Roman technical terms, historians' coined terms, and common modern English terms. Rome had controlled territories for centuries before Caesar. Heck, as soon as she exerted political influence beyond the immediate environs of the Tiber and Latium, she was technically embarking upon Imperialistic ventures. By the time of Caesar's consulship in 59 BCE, Rome administered provinces all over the Mediterranean, including Sicily, Spain, North Africa, southern Gaul, Greece, the Balkans, and the Near East map. These were the possessions of a political entity we would now call an "empire." When we say the "Roman Empire", we mean the city of Rome and its ruler(s) and all those territories directly controlled by them.
But we also talk about the "Imperial period," or "the Empire" as distinct from the Republic. The "Imperial period" specifically refers to Rome's time under the rule of an autocrat whom we now call "the Emperor." The first was Augustus (traditionally), not Caesar, though Augustus preferred to be called princeps ("first among equals"). For this reason, we modern historians often call the "Imperial" period the "Principate." They mean roughly the same thing. The word "empire" as it relates to control of territory is separate from our word "empire" as it pertains to the political organization of Rome's governance, which shifted from a narrow oligarchic Republic to a quasi-hereditary autocratic system upon the victory of Octavian (later Augustus) at the battle of Actium in 31 BCE. When he died in 14 CE, the power transitioned to his stepson Tiberius, and then from Tiberius to Gaius, and so on. This is "the Empire."
So: Rome had an "empire" before the "Imperial Period," and continued to have that "empire" down to the end of the Roman west. The word "empire" and "emperor" and "imperial" come from the Latin terms imperium ("power to command") and imperator (one who wields imperium). In the Republican period, generals had to be acclaimed imperator by their soldiers--it was an act of respect. So Cornelius Sulla's troops hailed him as imperator during his governorship in Cilicia, and Pompey was hailed imperator by his soldiers in North Africa. It became traditional for all the autocratic "emperors" after Augustus to take for themselves this title imperator, even though virtually none of them "earned" it from their troops. It was just one more honorific title which the ruler absorbed, and so it lost its special significance and became a generic term.
Alkiabiades has given you a good overview of where you might be confused, and the two answers in this thread from u/LegalAction (via a linked older thread) and u/winter0215 might also be of interest. The old Republican institutions stuck around: the Senate and tribunes, for example. So it's a little bit of an arbitrary decision to say the imperial period begins with Augustus (who as the linked answer notes actually described himself as saving the republic - whatever his actual goals and results, that was the public-facing message). I think most people are pretty content with applying the definition to Augustus' reign, but if it seems a little fuzzy to you that's because it is! Nobody ever formally dismantled the republic to give us a nice clean definition. But Augustus ruled a long time, consolidated a lot of power, and set the precedent. That lengthy reign is important too, a whole generation grew up not knowing any other ruler (he was no Queen Elizabeth in that regard but still pretty good for the first century AD!).
