If someone wrote a book about President Dwight David Eisenhower, they'd refer to the president as "Eisenhower," on second reference. This holds up for just about every American.
And there are some Roman figures for which this holds. Gaius Julius Caesar is often named, on second reference, "Caesar."
But Marcus Aurelius Antoninus is often, "Marcus" on second reference. Rarely is he referred to as "Antoninus" or "Aurelius". His brother by adoption and co-emperor, Lucius Verus, is usually called Verus on second reference.
Publius Ovidius Naso is known to us as "Ovid," which is a simplification of his nomen. Why use that instead of his prenomen or cognomen?
The repeated names objection:
I have a feeling that someone is going to say that these seemingly random choices are based on the fact that other parts of their names were used by famous ancestors/prior emperors, so historians drew on whatever part of the name would set the person apart.
But that hardly makes sense.
The Romans solved this problem themselves by using "the younger", and "the elder," as in Cato the Younger and Cato the Elder.
And today we easily set people apart with the same name with a middle name or initial. I.E., George W. Bush and George H.W. Bush, both called Bush on second reference.
So, why the random name choices?
In general modern practice simply copies the short-form names that were in use either in antiquity or in extant mediaeval manuscripts.
First, we can nearly always exclude the praenomen, because no one's going to use that any more than we'd refer to either of the George Bushes as 'George'. Nomen and Cognomen are the options. The only exception that occurs to me is the emperor now known as Gaius (a.k.a. Caligula), for reasons I'll mention below.
In several cases, practice has varied over time. It used to be a lot more common to refer to 'Naso', 'Maro', and 'Tullius/Tully' rather than 'Ovid', 'Vergil', and 'Cicero'. I'd say there's no lockstep in cases like that. And again, there's Gaius, as he's often called in academic writing, though 'Caligula' is still more popular.
Very many of the names that have been settled on were settled on in antiquity. When Livy refers to historical figures, he usually uses their nomen: other writers use the cognomen. So to take one example you mention, 'Fabius' is the name that Livy uses for him. Livy never even mentions the name 'Verrucosus': we only know his full name thanks to two isolated references in Cicero and Festus. Similarly, we refer to the 'Pisonian' conspiracy against Nero because that's the name that Tacitus uses for C. Calpurnius Piso.
And this goes for very nearly all Roman figures. If they're mentioned in Livy or Tacitus or Caesar, they'll be known by the names the Livy and Tacitus and Caesar use. If they're only mentioned elsewhere, there'll generally be more variation.
In a special set of cases, the short-form names we use are derived from mediaeval manuscripts: specifically, the emperors, who are known by the names used in the headings that appear in mediaeval copies of Suetonius and the Historia Augusta. (I'm not about to complain, personally, because if you show me a name like Tiberius Claudius Caesar Augustus Germanicus, I'm going to have to look him up to find out who that is.)
Many of these emperors used short-form names in their own time too: for example, on their coinage. There's even an interesting bit of numerology based on this in a Christian poem, Sibylline oracles 5, which lists off Rome's rulers but instead of naming them, gives them numbers based on the initial of their short-form name. So Julius Caesar is 'twice ten ... he will have his first letter from ten', because the initials of his name, ι and κ in Greek, are the Greek numerals for 10 and 20; Augustus is 1 (α = 1), Tiberius is 300 (τ = 300), and so on all the way to Trajan (= 300 again).
I said I'd come back to Gaius/Caligula. He's an interesting one for a couple of reasons. First, the most popularly used name nowadays, 'Caligula', wasn't used for him until over 300 years after his death. 1st-3rd century writers consistently refer to him as Gaius. Sibylline oracles 5 gives his number as 3 (γ = 3), not 20 (for κ). It's only when you get to Eutropius, in the late 300s, that we find him introducing him as 'Gaius Caesar, Caligula by nickname'; and it's even later, in Aurelius Victor and the Historia Augusta, that 'Caligula' becomes the primary name. While he was alive, the name 'Caligula' was only used when he was a toddler. This is why some academic writing uses the more 'authentic' name, Gaius.
The switch to 'Caligula' is primarily thanks to manuscripts of Suetonius. Suetonius himself consistently refers to him as 'Gaius', but the headings in mediaeval copies of Suetonius give the title of the biography as 'Caligula'. As an example here's the oldest extant manuscript of Suetonius (cod. Paris. lat. 6115, 9th century), with the heading at the bottom of the left-hand page; Suetonius' text begins at the top of the right-hand page, starting with the name as reported by Suetonius himself. Suetonius' own title for the biography was almost certainly Gaius; at some point between the 4th and 9th centuries it got changed.
Contemporaries of the emperor, Philo, Seneca, and Pliny -- refer to him as 'Gaius'. But I think it's overwhelmingly likely that the emperor himself preferred 'Germanicus' as a short-form name. That's the name that takes up the most space on his coins, and he renamed the month of September 'Germanicus'. However, his reign was short and as a result he never became more famous than his father, who in the popular mind remained the Germanicus -- to such an extent that we don't know Germanicus' full name: he's always just called 'Germanicus'. We can safely infer that it was to avoid confusion with the Germanicus that the popular version of the emperor's name became 'Gaius', probably after his assassination (but who knows? maybe even during his lifetime).
By the way, as to the 'Older' and 'Younger' distinctions you mention: I believe these aren't ancient, but I'm having difficulty tracking info down.
Edit: I now realise I didn't address one of the examples you mention, Marcus Aurelius Antoninus. That is an interesting case, probably at least as interesting as Gaius, but I haven't researched it: I may have time to come back to look into that, but not just now I'm afraid. The Historia Augusta calls him Marcus Antoninus; my first suspicion would be that the usage 'Marcus Aurelius' comes from an effort to distinguish him from all the other Antoninuses, especially Antoninus Pius, but perhaps also Marcus Aurelius' successors. It might also have something to do with manuscripts of the Meditations, but I can't check that just now.