I'm not sure where you heard that the Caesars were imperial heirs, unless you're referring to the practice during the Tetrarchy of referring to junior emperors as Caesars. But that was centuries later, after the imperial office had changed completely. The principes from Augustus until the third century are called Caesars in pretty much all our texts, not just the Bible, for the simple reason that they were Caesars. Beginning with Octavian's adoption into the Caesarean family the princeps was inseparable from the family of the Caesars. After Claudius the name was no longer strictly inherited, as Nero, though a member of the same family (like Tiberius, Caligula, and Claudius through the Claudian side), was not related through the right branch to inherit the Julian name. Other than Vitellius, the Roman principes always co-opted themselves into the Caesarian family, and the adoption of an heir involved his inclusion into the Caesarian family as well (maybe that's what you're talking about?). The Caesarian name was simultaneously an imperial title (remember that in the early days of the Principate there were no real imperial titles inherent to the body of the princips himself--princeps doesn't quite count, as it was more of an indication of social preeminence than actual political power, and was intended that way) and an actual family signifier, although this became less true over time until it ceased to be the case at all.

Also, the emperor to whom Christ refers in the Bible is Tiberius rather than Augustus. Tiberius bore the title as well but, according to Suetonius, did not personally use it. Augustus was never a normal mode of address towards an emperor during the Principate, at least not until much later. For one thing, not all emperors were Augusti. Instead our texts tend to call the emperors either by their names or simply by naming them Caesar, which can cause confusion for beginners at Latin