Part of the answer is that the ‘accepted’ name depends on who you ask. To take Augustus as an example, when he was adopted by (Gaius) Julius Caesar (the dictator, crosses the Rubicon, stabbed on the Ides of March, etc.) he changed his name from Gaius Octavius Thurinus (his birth name) to Gaius Julius Caesar Octavianus. Not long after this he dropped the ‘Octavianus’ bit to be known simply as Gaius Julius Caesar. This was so as to emphasise his connection with the famous Julius Caesar and cement his position in people’s minds as his rightful heir.

But Mark Antony saw himself as the natural heir to Caesar, and so he called the-boy-who-used-to-be-called-Gaius-Octavius not ‘Gaius Julius Caesar’, his preferred name, but simply ‘Octavian’. This emphasised Octavian’s natural father, Gaius Octavius, and distanced Octavian from his adoptive father, Julius Caesar. (In other words, what you called C. Julius Caesar/Octavian indicated whose side you were on.)

As evidence for this, we have a very interesting slingshot bullet from this period from the town of Perugia from the Antonine camp which has the words OCTAVI FELLAS on it (“For Octavian, who sucks d**k”).

Anyway, it seems that the only source that later Roman historians (e.g. Plutarch – not Roman but writing about Roman history) had for this period was sympathetic to Antony, not Octavian, and therefore refers to Octavian using Antony’s preferred term of ‘Octavian’ rather than Octavian’s preferred term of ‘Gaius Julius Caesar’. And this preference has been continued ever since. By calling him ‘Octavian’, we are repeating anti-Octavian sentiments from 2,000 years ago.

Indeed, if the sources for later Roman historians had been pro-Octavian instead of pro-Antony, the common, traditional, accepted name for Octavian in this period might well have been Gaius Julius Caesar. Who knows?

Later Emperors tended to have preferred and not-preferred names, too, and (modern) historians are now more aware of the bias implicit in particular names. When it comes to the Emperor Caligula, for example, the name ‘Caligula’ for example was only ever a nickname, but this was the name used by later historians, who were generally very negative about Caligula. Since these are the only sources that have survived, the name ‘Caligula’ stuck for a long time. Only relatively recently have modern historians begun to use the more neutral term ‘Gaius’ instead of ‘Caligula’.