which is definitely not how the Romans would have referred to them

This question is predicated on the belief I just quoted, and it's erroneous. In the majority of cases it is how the Romans would have referred to them, and appears typically to have been how the emperors referred to themselves. The use of single names instead of full or partial imperial titulature is as old as the Principate itself. Tacitus refers to Tiberius sometimes as Caesar, but just as frequently as Tiberius. Similarly he calls Nero Nero, Claudius Claudius, Vitellius Vitellius, Otho Otho, Vespasian Vespasian, and so on. One might try to say that that's because Tacitus is talking about multiple emperors, but it's not the case: in the Agricola there's only a single emperor, Domitian, and Tacitus calls him Domitian. He makes mention of Trajan and Nerva briefly in the same text, calling the former Nerva Trajanus and the latter Nerva Caesar. To Domitian's name he appends nothing. Similarly, Suetonius calls the emperors by name: Tiberius, Gaius, Claudius, etc. Pliny's letters to Trajan are addressed C. Plinius Traiano Imperatori, "C. Pliny to Trajan, Imperator," and Trajan's letters back are simply Traianus Plino, Trajan to Pliny.

Already by the first century AD the conventions of Roman Republican nomenclature were breaking down, and they were totally dead by the early second century. Factors included the complication of names due to extending citizenship throughout the empire (eventually resulting in names like Flavius and Aurelius becoming titles, not names in their own right) and to the diversity of ethnic and linguistic influences within the city and empire, and to the preservation of noble names in lines that had otherwise gone extinct, which began particularly early in the imperial family. Already in the Republic names were not necessarily as set in stone as we might wish them to be in our neat prosopographies. Cicero went by Marcus Tullius and Marcus Cicero both. And the reversal of names is quite common, wherein one element of the name (often the cognomen) is placed in front of another element (often the nomen, but sometimes even the praenomen) for no apparent reason. Cicero does it quite frequently, and we know in many of the cases that he preserves that the people to whom he refers are more typically called according to regular naming convention. And as names became more complicated typically certain elements would be selected. Pliny's name is one of the most famous examples. His full name is C. Plinius L.f. Oufentina Caecilius Secundus. L.f. and Oufentina are just indicators of his parents, but his name is really two names, Plinius and Caecilius Secundus, with two nomina--Pliny's name preserved the names of both branches of his family. Yet he is invariably just called Pliny in the texts. Why some elements might be chosen over others is not necessarily regular. A father might be called by a completely different name than his son, whose name nonetheless preserved the same elements of his father. Much of this must be chalked up to personal choice, or social convention--Romans had few qualms about using names and nicknames assigned by other people, and many cognomina continued even in the late Republic to be socially-assigned rather than hereditary, sometimes resulting in the accumulation of cognomina or the dropping of them between generations (e.g. the loss of Strabo from Pompey's father to Pompey himself) when they no longer fit. Thus clearly Trajan preferred to be called Trajan, whereas Constans II and Caligula are both nicknames that entered the general written convention within their own lifetimes, but that were probably not officially used by the emperors themselves.