In the early 20s BC, the system was most certainly still regarded as the Republic. In fact, the name for their system of government remained a 'Republic' until the reign of Diocletian when it became clear, even to the Romans, that it was clearly no longer a Republic. Of course the 'Republic' (or what we would call the early empire) changed drastically over time, but I digress. (I answered a question recently about if the third century empire still regarded itself as a Republic. tl;dr, yes, to a point: https://www.reddit.com/r/AskHistorians/comments/8wya2j/when_his_son_died_in_battle_trajan_decius_made_a/)

As for Augustus, after his first (27 BC) and second settlements (23 BC) with the senate, his powers were, in effect, a combination of the stations of consul (commander of legions), tribune and censor. But he was very concerned about how people saw him. He did not simply demand that his new... subjects refer to him as a dictator or king. These words had become anathema to the public. Instead he opted for, as you say, 'princeps' meaning 'first citizen.' This way, the facade of the Republic could be maintained in the eyes of the overwhelming masses while he retained his newly acquired powers.

Hope this helps.