I'm taking this text from the Wikipedia article. "The ancient Greeks used one of three words to refer to the home location of the Spartans. The first refers primarily to the main cluster of settlements in the valley of the Eurotas River: Sparta.[16] The second word was Lacedaemon (Λακεδαίμων);[17] this was also used sometimes as an adjective and is the name commonly used in the works of Homer and the historians Herodotus and Thucydides. Herodotus seems to denote by it the Mycenaean Greek citadel at Therapne, in contrast to the lower town of Sparta. It could be used synonymously with Sparta, but typically it was not. It denoted the terrain on which Sparta was situated.[18] In Homer it is typically combined with epithets of the countryside: wide, lovely, shining and most often hollow and broken (full of ravines).[19] The hollow suggests the Eurotas Valley. Sparta on the other hand is the country of lovely women, a people epithet"
It didn't have that many...
The name of the political community and territory was Lakedaimon. As the wiki page you're citing says, this was true as early as the earliest works of Greek literature. The people of this territory and state were normally called Lakedaimonians. The other names are rare.
The name Sparta refers mainly to the cluster of villages that formed the "urban" core of the state. Lakedaimon was not very urbanised compared to other Greek regions, and it wasn't until the Hellenistic period that the towns of Sparta coalesced into a single fortified city. Unlike other Greek urban centres, Sparta did not give its name to the people who lived there; as I said, the common name for the people of this state was Lakedaimonians. However, Greek sources will sometimes use the word Spartiates (Spartiatai) to distinguish the full Lakedaimonian citizens (who were legally obliged to live in the towns of Sparta) from the disenfranchised groups they ruled over. For example, the Lakedaimonian contingent at the battle of Plataiai consisted of 5,000 Spartiates, 5,000 perioikoi and 35,000 helots.
Finally, there is Lakonia. This is the proper name of the valley of the Eurotas - the regional term for the core agricultural land dominated by the towns of Sparta and by the community of the Lakedaimonians. During the Archaic period, the Lakedaimonians subjected many neighbours, extending their domain across half of the Peloponnese. However, by the end of the 4th century BC they had lost all the territories beyond Lakonia itself, and so authors like Aristotle could use Lakonia and Lakedaimon interchangeably.
There is some overlap and confusing use of these terms, because the Greeks didn't tend to use words as precisely as we would have liked them to. But generally there's not really that much variation. The region is always known as Lakonia or Lakedaimon. The people are almost invariably known as Lakedaimonians. It's probably more our own blanket use of "Sparta" and "Spartans" that causes confusion when we are confronted with actual Greek practice.