Why Did Augustus Rule A Republic?

An empire, by definition, is“ a group of regions that are controlled by one ruler or one government; especially : a group of countries ruled by an emperor or empress ”. In contrast, a republic is “ a country that is governed by elected representatives and by an elected leader (such as a president) rather than by a king or queen “. The difference between these two is where the power to rule the country lies, whether it be an individual or many elected officials. Using these definitions it would be easy to make an argument that the Roman Republic was void much earlier than Octavian's transition to the first emperor Augustus, which is what textbooks and many historians will use to mark its end. There were several occasions in the Republic's history where power was transferred, or taken by force, to a single man for longer than the allowed office of dictator. However, until Augustus’ reign these dictatorships lasted for a time and then democracy resumed soon after. I believe that for the Republic to truly end, there must be more than one emperor in a row, rather than immediately returning to the way it was before.
After his victory over Pontus he returned to Rome without any political position, but his trained army behind him. He used the troops to take power and the office of dictator, creating new laws that allowed him to eliminate all political opposition or people he otherwise didn't want around. This certainly seems like the end of democracy in Rome and the beginning of an empire, but Sulla’s rule only lasted three to four years, after which the senate took power once