So, I'm going to largely discard the Roman foundation myths since there are several and they aren't exactly compatible with one another. Out of them, the one I feel holds the most weight is likely that Rome itself was founded by colonists from Alba Longa, but I don't believe we have any concrete proof of that.

Prior to Rome itself existing, the area in Rome-proper was various Latin settlements (or at least settlements that were part of the Latial culture) on the various hills. Eventually, they ended up growing and merging together, with the various tribes themselves becoming the founding tribes of Rome.

The region that Rome is in is Latium, which is the land of the Latins. The actual etymology of "Latin" is contested, but is generally believed to come via Italic *tlatos or *tlatus, both having the same PIE root, but meaning 'wide' or 'side' respectively, referring to the geography of the area. The Latins were a subset of the Italic tribes, likely the second immigrating group of them to the peninsula. They were closely related to the Falisci (the Italic tribes around Falerii in southern Etruria), and more distantly related to the Oscans and Umbrians in the area, such as the Samnites and Sabines. One thing to note is generally when you see names of Faliscans/Samnites/Sabines, they are Latin versions of the names - for instance, the Samnite Gaius Pontius most definitely never was called that in Oscan, being closer to Gaies Ponteis or so.

The Latin tribes initially were part of a larger tribal confederation of sorts that we coin the Latin League, of which Rome was a member and later the leader of, until the Roman Republic dissolved it after the Second Latin War in BCE 338. Prior to this, significant parts of Latium, including Rome, were under Etruscan influence from the north. The Etruscans were not an Italic or even an Indo-European peoples. They were likely related to the Rhaetians and Lemnians, and referred to themselves as Rasenna. By BCE 264, the Roman Republic had conquered Etruria, and they began to be assimilated, though the language itself persisted until early in the common era.

Italy itself, if we're referring to the peninsula, was settled by various Italic tribes which had emigrated there in at least two discrete migrations (the Oscans and Umbrians first, and then the Latins and Faliscans later), the Etruscans, Greeks in Magna Graecia, Gaulish (Celtic) tribes in Liguria in the north-west, and various indigenous peoples (Pre-Indo-European). No one group in particular 'claimed' Italy. There were various tribes, tribal confederations, city-states, and occasionally petty kingdoms on the peninsula. Rome was (likely) initially a tribal kingdom within the Etruscan sphere of influence, with their final legendary king being Lucius Tarquinius Superbus, himself being of the gens Tarquinii and thus being Etruscan from the Etruscan city of Tarchuna (Tarquinii), and supposedly escaping to their and raising Etruscan forces to try to retake Rome. The Latins, as said, were loosely organized into the Latin League, the Etruscans were loosely organized into the Etruscan League or "League of Twelve Cities", and other tribes such as the Samnites are noted to be loosely organized as well, being able to organize under a single leader when called upon such as in the Samnite Wars. There were also the Greek city-states in the south such as Syracuse.

If you go back further, it gets much murkier. Prior to Greek settlement in the 700s BCE, the region would have been a mixture of Italic tribes and Pre-Indo-European tribes, and possibly other unknown peoples, and possibly unknown Indo-European peoples. The Etruscans were in the area at least from BCE ~900 on as the Villanovan culture, though their own legends state that they came from elsewhere. The Villanovan culture likely isn't their native culture as it is a development of the Urnfield culture, which is Indo-European, whereas the Etruscans were very much not. They likely adopted many cultural elements of the Italic tribes who had emigrated to the peninsula prior to them.

There is no known single name for the peninsula before the founding of Roma. Italy at the time of Rome's founding was settled by a number of tribes, none of which was dominant, and most of whom were Italic speaking, although the Etruscans, who would come to dominate Italy in the 6th century BC, did not speak an Indo-European language at all. "Italia" as a name for the Italian peninsula dates to after the founding of Rome and is Greek in origin.

Here's a decent timeline: https://www.historyfiles.co.uk/KingListsEurope/ItalyKingdom.htm