At least in the German speaking area there has to be made an important distinction between Spartans and Lacedemonians if one wants to be historically accurate.

"Spartan" is correctly used only for full citizens of Sparta. So prototypical for sons (daughters) of two citizens. They were the ruling, but still a quite small and elitist social group in the area of Laconia.

"Lacedemonians" was already used by ancient authors to name the inhabitants of all of Laconia. So not only the first social class of people living there, the Spartans, but also the Perioeci, who where tolerated population that has been living there for a long time but were not citizens of Sparta and partly even the Helots who you could describe as the "stateslaves" of Sparta.

Athens is a different kind of animal in my opinion because the city itself was such an overwhelming center of the ancient world. That's a big contrast to Sparta, which precisely consisted of five villages who had different names themselves. So calling the "area" of Sparta "Laconia" and it's inhabitants "Lacedaimonians" is far more accurate than calling "Athenians" "Atticans".

Although talking about ancient terminology is a minefield I hope this answer helped.