So, there are two "Italies" to work with here. There's a geographical space "Italy" and a political space "Italy." The two develop sort of concurrently, but we should be clear that the political and geographical spaces are different until the end of the process.

"Italy" the geographical space occurs in the 5th century historian Herodotus, who used the term to describe only the Greek states located around the Gulf of Taranto. Other parts of Italy had different names, Oenotria, Tyrrhenia, etc. The name is older than Herodotus, of course; that's just the first time it shows up in the surviving sources. The 4th century historian Antiochus applied the name to a larger area, from Naples south (so the Boot). Eventually the term came to be used for the entire peninsula below the Arno-Rubicon line, and stayed there until probably Octavian, Antony, and the assassins promoted Cisapline Gaul from a province to Italy proper as part of the ratification of Caesar's acts, though that might have happened when Octavian (as Augustus) divided the administration of Italy into regions later in his reign. There's a full bibliography on the geographical use of the word "Italy" in the Inventory of Classical Greek Poleis under the chapter on Italy if you want scholarship on this.

That doesn't mean Cisalpine Gaul was excluded conceptually from Italy. Cato the Elder referred to the Alps as "the shield of Italy" which suggests a close relationship between the mountains and the peninsula. Some scholars think this means Cisalpine Gaul was at one point included in "Italy" and was later reduced to a province, a reduction in status repaired under Augustus, but others point out that being the shield of a thing does not mean you're the thing itself. Williams, Beyond the Rubicon is the book to find out about the relationship of Cisalpine Gaul to Italy.

But the important thing to take from Cato here is that, whether Cisalpine Gaul was formally included in the geography of "Italy" or not, it is playing a role in the political space "Italy" - being a shield is a military application, and Cato is thinking of Italy as a thing that can be defended. When that happened isn't clear at all. I've seen three points: the Pyrrhic War of 280-275, the Celtic invasion of 222, and the Social War of 91-88.

The argument for the Pyrrhic War is that the settlement of the war saw Rome establish a sphere of influence over the Italian peninsula and the various states recognized Roman hegemony. The case for 222 is that the Celtic invasion defeated at Clastidium caused Italian cities to recognize each other as "a community of mutual interest"; whatever problems they had with each other was peanuts compared to their problems with marauding Gauls. The case for the Social War is pretty straight forward. A chunk of Rome's allies (in south-central Italy) for reasons that aren't entirely clear (either because they wanted Roman citizenship, or because they wanted independence, or because they wanted a federal Italian state, or maybe the revolt started for none of these reasons and they only adopted these goals after the revolt) broke from Rome and fought a very nasty war. The rebels founded a federal (or confederate, depending on what you think they were up to) capital named Italia or Italica at Corfinium, and struck coinage with the legend "Italia" in Latin and Oscan. Pobjoy in an article "The First Italia" argued that this was really a unitary state in a real sense.

There are problems with all of these positions. Neither the Pyrrhic War or the Celtic invasion, if they did create a Roman sphere of influence or a "community of mutual interest", whatever that is, had enough effect to prevent Italian allies from defecting to Hannibal after 216, and the slogan Italia in the Social War (largely fought by the same groups as defected to Hannibal) was not attractive enough to maintain the alliance against Roman grants of citizenship in 90 and 89.

"Italia" as a slogan surfaced again in the late 1st century. Augustus claims to have received a loyalty oath from "tota Italia" - all Italy - in preparation for the war against Antony and Cleopatra. It's not clear what that means exactly, whether it was personal (i.e. each individual personally swearing loyalty) or communal, and if it really was all Italy (Syme thought the cities that were clients of Antony must have been excluded from that oath). In any case, that war was conceptualized as a war of Italy against the East. Augustus also established a lot of institutional changes that helped develop Italy as a political community, most notably the establishment of veteran colonies both inside and outside Italy, and the administration and maintenance of the roads. A lot of what happens under Augustus looks like the kind of thing we include in "nation-state formation" when we're talking about the modern period.

[Edit: I was wrong about Italiote. It's Italic. Italiote refers to the Greek speaking population of Italy] As for why the Celts of north Italy aren't considered "Italian", that's simple. Greeks weren't either, because neither group spoke an Italic language. There's a family of languages like Oscan that are closely related to Latin (Oscan is close enough that sometimes it looks like people just didn't know how to spell), and the people speaking those languages are the ones we call Italian, when we aren't speaking geographically.