The Greek colonizing horizon begins about the middle of the 8th century, with the first "Greeks in the West" at Pithekoussai in the Bay of Naples by 750 BCE, Neapolis by 700, and several sites on Sicily 740-680 BCE (or so), including Syracusa and Megara Hyblaia. A lot has been written on the identities of these peoples. This is a good summary of the basics of the very complicated discussion. For our purposes, we can't really say much about this early period from literary sources. They are all from much later and are colored by their understanding of Greek society in their own time. From my own work in archaeology, I can tell you that in these very early "colonies" of the 8th and 7th centuries, the inhabitants did typically "Greek" things, but not quite the same way as they had done them back home. They made Greek pottery, but they experimented with new shapes and new styles of decoration. They buried their dead in a "Greek" way, but with variations. They worshipped Greek gods and built Greek sanctuaries. They spoke Greek dialects. They were, essentially, Greeks. One of the earliest examples of written Greek comes not from the Greek "heartland," but from Pithekoussai, way over in the Bay of Naples. It is a Greek drinking vessel, a kotyle, all the rage at this time, decorated in an "orientalising" (stylish) Geometric style, and written on it is "I am the cup of Nestor" (a well-known character from the Greek Trojan War cycle). This cup was being made when the stories which would become Iliad and Odyssey were just beginning to coalesce and perhaps be written down in something like their "final" form for the first time. It was being created when the concept of a "polis" was just beginning in Attica, the Argolid, and the Korinthia. It is, in essence, a quintessentially Iron Age/Geometric Greek object. And it was found in Pithekoussai.

If we fast-forward to the Archaic period and down to the end of the 6th century, many these places you have mentioned are thriving and they do consider themselves Greeks, just as much as the people of Miletus do (on the coast of Turkey and recently under the thumb of Persian overseers). They speak Greek, they are actors in a Greek Mediterranean economy which connects them culturally and artistically, and, most importantly, they are fervent participants in the Pan-Hellenic games, especially at Olympia and Delphi. They send contingents to these events, they compete, and the elites of Massalia and Syracusa and Tripolitania rub shoulders with, drink with, sacrifice with, argue with, dedicate with the elites of Athens, Korinth, Thebes, Sparta. The games give the denizens of the far-flung enclaves a central venue for proclaiming their Greekness, and they leave treasuries at Olympia and Delphi which are decorated in the most expensive and cutting-edge fashion, telling familiar Greek myths with their sculpture, and containing the rich dedications of their poleis.

A great basic source for reading about the Archaic period of Greece is Osborne, Greece in the Making, which is an excellent book all around, and has many discussions of Greek colonisation throughout. He talks about colonies, what "colony" means in this context, and also about panhellenic participation and what that all means both early on and later.