There are several fairly distinct concepts to parse in your question. I'll start by examining the emergence of national identity in Italy and Sicily, spend a little time on the South's role within the wider narrative of united Italy in two critical phases (immediately after unification, and immediately after the Second World War, the former in more political terms, the latter in more social terms) and subsequently dive into a discussion of more recent topics linked to regional identities and regionalism in Italy as they were instrumentalized in the early 90s. If that seems like a lot to read through I completely understand, you can read the next two paragraphs and then skip directly to the last part (three replies down).

To answer your first question right away, the term "Sicilians" always exclusively refers to people self-identifying or identified by others as being from the actual island of Sicily (and usually minor outlying islands in the vicinity of Sicily). "Southern Italian" instead refers to a geographic designation which includes people from all of the South of Italy, of which Sicily is an important and distinct part. The reason "Sicilian" is never applied as a general appellative for people from the mainland possessions of the Kingdom of Sicily (and later, the Kingdom of Two Sicilies) is the same reason why there is very little to show for any sort of historic notions of identity in Italy beyond the city or town level: the Italian unification movement overlaps with the very emergence of nationalism as a concept (in Italy, but also pretty much in the rest of Europe too). The Kingdom of Sicily, Kingdom of Naples, Kingdom of the Two Sicilies, and the various other permutations of government in Southern Italy that existed prior to unification had a very weak concept of nationalism as we would understand it today. The same was true for much of the rest of Italy: Prior to the 19th century, most people would have identified themselves along the lines of their city or town (this was even true of citizens of the large republics spanning multiple cities, like Venice or Genoa, which to their end remained politically organized along concepts of subordination to a dominant city) although it's also important to point out that identity was seldom a clear-cut thing: many people also self-identified along religious lines, such through their parish or through their religious group, while others self-identified in social or economic terms, through their profession or guild. While you might find aristocratic people or the high bourgeoisie who would under some circumstances do self-identify as "Italian" (or from a specific region of Italy) especially when abstracting their place in relation to foreigners (or when abstracting the whole peninsula) there was no real "Popular" nationalist narrative present in Italy prior to the Napolenic period. And when a widely-accepted nationalist narrative finally did develop in Italy, it was by and large focused on the unification of the whole peninsula and its outlying islands.

This doesn't mean that all subsequent notions of identity were exclusively tied to the narrative of unification: as we will see below, there very much was a consciousness in political and intellectual circles of the uniqueness of the South (as there was for the other regions of Italy, but the south does develop a collective narrative much earlier than the North does, as we will see below). For some intellectual leaders and certainly for many individuals, there was undoubtedly a certain element of pride would develop in the unique history of the South, just as there were concerns for the south's unique problems. But the point is that this is a modern phenomenon, and prior to unity Sicilian identity just did not have the conditions to become some sort of pan-southern identity.

While the seeds of the industrial era, the seeds of Italian unification, and the seeds of nationalism are more or less contemporaneous, what does and does not get tossed into the bubbling cauldron from which nationalist narratives emerged in 19th century Europe can admittedly be subject to some interpretation. u/yodatsracist discussed some of the approaches to take when pinpointing what precisely constitutes "National Identity" in this answer from early in the days of r/AskHistorians which you might be interested in. I also wrote this short answer to a recent question on National Identity in Italy (where I also cite yodatsracist 's answer) and offer some more links to a few long narrations of mine on the slow trudge to the emergence of an Italian national identity. You might also be interested in this other answer as well as this answer to explicit questions on Italian Unification and national identity.

Linking all of the above with your other questions, I'll admit that while I also recall writing an answer on the Italian nationalist currents which existed specifically in Sicily I can't seem to find that answer anywhere. Summarizing here, the important take-away that ties into the broader trends outlined above is that Italian Unification didn't happen in a vacuum, and Sicily was no exception: after the Napoleonic era, nationalist activism existed in pretty much every single major Italian city (and lots of smaller cities too) and in Sicily (as elsewhere) periodic revolts challenged the restored monarchy's legitimacy. Indeed, the pan-European revolts which erupted in 1848 took on markedly nationalist narrative in Italy, and Sicily was actually one of the places where they were most organized and garnered the most support among the local intellectual class: Organizers coalesced in Palermo, where the local bourgeoisie attempted to reinstate the ancient Sicilian Parliament. While the revolt was ultimately squashed, a testament to revolt's broad participation is the multitude of currents and preferences which were expressed in the short-lived Parliament: appeals were made to the King Ferdinand II to either grant the island full independence, or at least construct a personal union separating the crowns in Naples and Palermo. But more so than explicit local nationalism, such was the revolutionary energy which fomented pan-Italian activism that some or other form Italian unification already seemed inevitable: in the very words agreed upon by the parliament, the proposed Sicilian constitution would have the objective to, "Lay the foundations to make Sicily a sovereign state, free and independent, and prepare her to become a member of a future Italian federation" (translation my own).

I move on to the unique consequences and mechanics of Sicily and the South's integration during unification below after the jump, which hopefully offers more color into the social and political dialogue which emerged during and immediately after Sicily and the South's integration.