Hellenistic Syracuse had about as large a population as Attica or Lacedaimon, so in my opinion it is large enough to warrant its own culture. But I agree that not all Sicilian Greeks should be Syracusan culture.
What you say is right, infact the term for all the sicilian greeks was "Siceliotes" (Sikeliotai), and the name for the greeks in the italian peninsula was "Italiotes" (Italiotai), Infact there are many "levels" of identity. There is the identity as "Polis", the identity as "ethnos" (dorian, etc...), the geographical identity (We said Italiotes, Siceliotes, and also the Thessalian have their own charateristics for example), and many other.
The Siceliotes where somewhat doric, itaiotes were more diverse (With a lot of famous infighting) but at some point they add also an
Italiote League opposed to Syracuse. Talking about Syracuse, we had at some point also the ambition of Sicilian kindom with
Agathocles. Phyrrus was called in Sicily as ruler becuase was is son-in-law.
Perhaps Trinacrian, derived from one of the Ancient Greek names for Sicily, could be a suitable name for Greek Sicilian culture.
Indeed we don't need to made up names, because it exist and its "Siceliotes"
However I'll expand the information on that, Sicily was divided in five populations at time.
1) Siceliotes,
Sikeliotai (Greek Colonies, this is your culture name for Sicilian greeks)
2) Sicels,
Sikeloi (italic indegenous people, heavily hellenized, the nearest to the eastern coast of sicily)
3) Sican,
Sikanoi (oldest indigenenous people, in the middle of sicily, their origin it's more problematic and probably not italic, they were hellenized too)
4) Elymi,
Elymoi (indigeneous people influenced both by pheonicians and by greeks, usually allied with Cartage, allied with Rome on the end, they claimed their origin from Troy and following some Myth also from Eracles)
5) Phoenicians (Carthaginian holding cities, fortress and trade posts)
All the indigeneous people at some point received (as many others) and high hellenic influence in their culture, we have many example in architecture, but mantained their identity somewhat at least until some point (but probably even more), infact the Sicels, when heavily hellenized opposed the Siceliotes with
Ducetious
This is to say that especially in the middle of Sicily a lot of cities (I don't know what cities they putted in, but there are a lot of cities there) are not even founded by Greeks or Phoenicians, but by indegeneous people strongly hellenized at some point.
In this map you can see the Island before the greek arrival, many of these cities were founded by indegenous people, others were instead founded by greeks but in places already controlled by others. For example it's the case of "Lentini"
It's surprising but the english wikipedia it's very poor on these arguments, off course italian wiki it's more filled about that. I own also some book on the argument when I studied greek history at university, but it's in Italian, so for now I have no better english links for who is more curious.