Over the course of a few hundred years, certain names tend to go out of favor and others into favor - and of course you constantly have foreign names being introduced, especially in times of invasion. But, as with all of the other Romance languages - the local dialect of Latin developed into its own language, Italian. Vincentius became Vincenzo, Flavius became Flavio, Julius became Julio, and so forth. The two names you gave as Roman names were not very common in Rome: Spartacus was likely a Romanized Thracian name and Romulus was coined in reverse, sort of, by adding the diminutive suffix (-ulus) to "Rome", to mean "little Rome".

The slow introduction of Christian names is another thing that had a huge effect there - with Giuseppe/José/Joseph and John/Jean/Johan/Iohannis etc being localized versions of Hebrew names found in the Bible. This is the origin of a huge number of names in every European language - maybe a majority.

I recommend that you try searching for names you're interested in on Wiktionary, which often lists the etymologies of these names.

Roman culture became modern Italian culture through a gradual process of evolution, there is no one definitive moment you can point to. We're talking about 2000 years here, practically anything can happen linguistically in such a timespan. And drawing a distinction between Romans and Italians is inherently artificial. There were Italians, as in residents of Rome's Italia province, centuries before the fall of the western Roman Empire, and many Italians still considered themselves Roman long after it did. There is no mutual exclusivity.

At any rate, I'll give you the etymology of the the names you mentioned. Giuseppe comes from Hebrew, it's a mutation of "Joseph", the father of Jesus. The shift to Christianity is responsible for it. Vicenzo came from the Roman surname "Vincentius", just some minor shifts in word endings. Julius became Giulio, Romulus became Romolo, and Spartacus was a Thracian who never had a Roman name at all.

EDIT: Eh, started writing this before Pugy'a post. Apparently Spartacus was a latinization of his Thracian name.

It seems like you're asking about how Latin became Italian, but one might just as well ask how it became French, Spanish, Portuguese, Catalan or Romanian. I therefore cordially suggest that you may have better luck getting an answer to your question over at /r/linguistics where there are lots of smart and learned people who would like nothing better than to explain to you, at great length, how Latin begat its various descendants.

The context of the opening post is a bit different from what I first expected when reading the title; consequently, I think that there's another relevant question here.

Was there ever in time a major shift when people located on the Italian peninsula progressively began self-identifying moreso as Italians as opposed to Romans?

Well, it had to do with the fall the Roman Empire, as after that the Italian peninsula was invaded by various ethnic groups who then ruled over the area, any sort of national/ethnic idea/unity wasn't very popular.

This coupled with the fact that "Roman" was later used as a citizenship title in the later years of the empire than an ethnic one. Once the empire was gone, people were more concerned (like others posted) with their village/duchy/kingdom rather than some homogeneous ethnic group. It would take a couple thousand years before guys thought to make a country out of them. A lot can change in 1400 years! Also, people don't know how to spell their own name if they can't write or read, thus it changes over the years.