In general, from what I understand to be true, I agree with you.
In practicality, Gaius Julius Caesar was indeed the first de facto emperor of Rome, but in the eyes of the Roman law, he was still a dictator (albeit dictator perpetuo - dictator in perpetuity). He held the military power as dictator (and in the eyes of Roman law, when there was a dictator, he was the principal consul which wielded all imperium on behalf of the Senate), he held all political power (mainly through intimidation and killing off his opponents, albeit not through Roman law), and he held all religious power (he was the Pontifex Maximus). Ergo, Caesar was in practice - but not in law - the first Roman emperor. Unfortunately for him, the Senate didn't agree with this, and consequently the Senate gave Caesar the boot.
Octavian (later Augustus) was the first Roman emperor (actually called a princeps, from which the English word prince descends) in the eyes of Roman law. He had all military power, all political power, and all religious power. He was given the power of imperium throughout all Roman lands by the Senate. But I think what seems to be overlooked is that Augustus weilded this power under the intentional guise presented to the Roman people that the Republic was still in existence. Augustus was smart enough to let the Senate wield power, but he made them ever mindful that at the end of the day it was really the princeps that wielded the real power. So long as the aristocrats and powerful plebs could advance through office, there was less pressure to do away with the princeps. Additionally, Augustus had his power of imperium extended by the Senate every 10 years - a move which made the Senate and people think that Republican principles were still in place... he didn't have the power of imperium in perpetuity.
The difference between Augustus and Julius Caesar is that Augustus succeeded in playing his hand carefully exactly where Caesar failed. Caesar basically rubbed people the wrong way, and despite his outwardly altruistic expressions of foregiveness, people didn't believe him. For too long the Roman Senate felt that Caesar was illegally waging war in Gaul and was trying to get at ultimate power, and when he crossed into Italty and triggered the civil war, his fate was sealed - Caesar was doomed to die.
Augustus, on the other hand, played the game of patience with the Senate, made them feel that he respected the old ways of the Republic, and he made gestures to make influential people believe he didn't have the same lust for power that Julius Caesar did. At one time, when he was offered ultimate power by the Senate, he refused it once - realizing that he was the only person who could help Rome through the turbulent times after the Civil War, Augustus knew they'd insist he accept ultimate power, and he did. And when he had it, he didn't make the Senate think that he wanted to become the new rex Romana (Roman king).
Caesar didn't address the problem of succession; and familial succession - in addition to ultimate control of all power over the state - is the other factor that defines a monarchy/empire. Caesar left no clear heir to his power, and even if he had, the Senate would have rejected it, because only a king could appoint an heir without the consent of the Senate (apparently even the Roman kings - before the Republic - required approval of the people). Augustus tactfully took care of the issue of succession by ensuring his desired heirs to power were appointed to lesser positions in power that wouldn't worry the Senate, but would allow the heir(s) to be groomed for leadership in the future. Augustus used the system to control the system. Caesar used the sword to control the system.
In short, Caesar was the greater imperator, but Augustus was the greater politician. Caesar was doomed to fail by his own arrogance - Augustus was doomed to succeed because he didn't let his arrogance show. Augustus succeeded in walking the tightrope from the time he became the undisputed military champion of the Republic until he died in 14 AD. Augustus was a smart dude - boring, but smart. Caesar was the star that shone twice as bright, but burned half as long.