To my (sadly limited) understanding of Roman history, there wasn't a precise cut-off date of the Roman Republic following Augustus' ascension to power in the 1st century BC, and the subsequent 'emperors' maintained (or at least tried to maintain) a general veneer of continuation following the footsteps of the old republican traditions. This continued for three-or-so more centuries until Diocletian (kinda-sorta) did away with the façade and formally dressed the Roman state in more autocratic clothing.
Now my question is; is this view of history correct? If not, is there any other points in history in which 'the Republic' could have arguably became 'the Empire'?
Moreover; is there any point in time in which the people living in Rome realised the transition of power from the Senate to the 'Emperor'?
Per the Romans of the time such as Cicero, the republic died once Julius Caesar won the civil war. There was an attempt is resurrecting the republic with the assassination of Caesar, but really the republic died with Caesar and the institutional emperorship was concentrated and formalized with Augustus.
The fact that a boy of 19 could come to such a prominent political position simply by being adopted by Caesar after his death really illustrates the point. Prominent citizens have died and left young successors in their lieu (Marc Anthony is one example) but no one before Augustus gained that level of political leverage and position based on the death of one man.
I think that after Augustus came to power that is often considered to be the (unofficial) end of the republic and begining of the Roman empire. Augustus called himself first among the citizens and refused the title of emperor but it seems clear he was emperor in most of the ways that matter, and had more power as an individual than the senate did.