Every text I have come across uses Lacedaemonia and Sparta interchangeably; the same with Lacedaemonian and Spartan. But it is clear, for me at least, that the name the ancients used were Lacedaemonia and Lacedaemonian. So my questions, if my premise is correct (otherwise please note so), are:
This isn't my field, but I found the following in Nigel M. Kennell, Spartans: A New History (2011) pp. 14-15.
There is a welter of names -- Sparta, Spartans, Spartiatai, Laconia, Laconians, Lacedaemonia, and Lacedaemonians -- each with a slightly different connotation and history. In the Classical period, the southern Peloponnese under Spartan control was commonly called hê Lacônikê (probably gê), "the Laconian (land)" (Hdt. 1.69.4; Thuc. 5.34.1). The territory we call Laconia was also referred to as Lakedaimôn. Unfortunately for clarity, the city of Sparta was also called Lakedaimôn, while the official designation of the Spartan state was hoi Lakedaimonioi, "the Lacedaemonians." ... From time to time, the ambiguous designations "Laconian" (Lakôn) also crops up (e.g. Hdt. 1.68.2, 8.2.2; Thuc. 3.5.2). Thus, "Lacedaemonia" might designate either the Spartan civic center or all of the Spartan territory, and "Lacedaemonians" could be Spartan citizens, the non-Spartan perioikoi, or a combination of the two. "Proper" Spartans, those adult males who maintained their commitments to the state, were Spartiatai, "Spartiates." Unlike Lakedaimôn, which has resisted etymological explanation, Spartê (Doric Sparta), is generally agreed to be connected with the verb speiro ("I sow") and mean something like "the sown (land)" -- a suitable name for a newly founded community. The name is appropriate, as there is no archaeological evidence for settlement on the site before the early Iron age... [In the early 1990s] excavations on the acropolis of Thebes in Boeotia uncovered a cache of clay tablets from the Mycenaean period, the Late Bronze Age (roughly 1400-1100 B.C.E.), written in Linear B. Several of these tablets mention men called either "the Lacedaemonian"... or "the son of the Lacedaemonian"... who may have played some role in cult activity. From the tablets, we know that Lakedaimôn is the oldest geographical designation, dating back to the second half of the second millennium B.C.E., which inhabitants of Sparta may later have adopted in order to project an image of themselves as guardians of the old Lacedaemonian heritage, a process underway by the eighth century. (pp 14-15)
So to answer your question at least in part, it seems the idea of "changing" names is inherent in the name itself, and even the Spartans themselves seem to have taken up this practice. Any definition has to be arbitrarily imposed by historians, guided but never dictated by the objects of study, and in this sense there's no name that has an authoritative claim to being the singular, most accurate, truest name. Personally I would be interested to know when people began to refer to the region by the name of its most powerful city, as this might reflect something about our view of history and the world, but again, this isn't my field at all and I have no familiarity with the historiography. In the end, though, I think you've arrived at this question with the idea that there is a consensus on naming places, which isn't even true today--we refer to countries and peoples differently than they refer to themselves and their own countries. It may be that your authors are interchanging the two names as a way of referring to different communities, rather than geographic locations, which existed in the same space but did not see themselves as equal inhabitants thereof.
Hopefully this was insightful, and perhaps someone can elaborate on the historiography.
According to Graham Shipley (using both the Laconian Land Survey and the Copenhagen Polis Centre Project), Lacedaemonian and Spartan should not be used interchangeably. Though all Spartans were Lacedaemonians not all Lacedaemonians were Spartans. The Spartans were those that had full political rights (as discussed by Aristotle in his account of the collapse of Spartan manpower - Pol.1270a15–1270b6). Those that didn't have full rights but were still citizens were the Perioikoi. They were members (though not equal members) of the Lacedaemonian polity.and lived scattered through Laconia and Messenia in 20 or so Poleis (in the sense of a town or community) that formed a single Polis (in the sense of a state).
However as with all things Spartan there are complications because outsiders (ancient and modern) never really understood the society as it existed. So for example Xenophon will use the term Spartan to sometimes talk of those with full citizen rights (such as casualties at the battle of Leuktra) but also sometimes for those without full citizen rights (such as when he talks about the mercenary Drakontios - An.4.8.25). As a result the terminology can be slippery and inaccurate when used by these outsiders.
Shipley (2006) ‘Sparta and its perioikic neighbours: a century of reassessment’, Hermathena: 51–82