In the Classical period, there is only one name for the Greeks: hoi Hellênes, the people of Hellas.

There were, however, numerous ways to refer to the Greeks in earlier times. In the works of Homer, the most common name for the Greeks is hoi Akhaioi, the Achaeans; this may have been an archaising way to refer to the Greeks, which originally dates back to the Mycenaean period. In Classical times, the name referred only to the people of Achaea, an unremarkable region in the north of the Peloponnese.

Additional names found in the Iliad and Odyssey are hoi Danaoi, the Danaans, and hoi Argeioi, the Argives. Like "Achaeans", the latter is a pars pro toto elision - the name of one part is used as a name for the whole. The city of Argos was one of the largest and most prominent cities in mainland Greece, and so its people were apparently used as a shorthand to mean all the Greeks.

None of the Homeric names remained in widespread use. For most of Antiquity, the standard Greek name for the Greeks was hoi Hellênes. However, the Greek world consisted of many hundreds of smaller states, regions, and dialect groups, all of which had their own names (and sometimes several names). Your example, "Lacedaemonians", does not mean "the Greeks" as a whole - it is the name for the inhabitants of Lacedaemon, a region in the southeastern Peloponnese of which the main urban centre was the town of Sparta.

There are certainly many names for the full citizens of Sparta. As citizens they were known as Equals and as Spartiates; as inhabitants of their region they were called Spartans and Laconians and Lacedaemonians; as speakers of Doric Greek, they were called Dorians; and finally, as inhabitants of Hellas, they were called Hellenes. But only that final word described the ancient Greeks as a whole.