Lacedaemonia actally was the entire region, which included cities like Tegea. Sparta was a single city and it woudl never be correct to call Sparta 'Lacedaemonia', althought the spartans were referred to as 'Lacedaemonian' quite often, because those things which set the spartans apart (stoicism, loyalty, minimalism) were also qualities of other people in the region, such as the Tegeans. SParta technically had no authority over other cities however, and in fact, Tegea was a common bitter rival, except those times they came together out of necessity, to fight greater powers, such as Athens.
The design of the phalanx sheild was attributed by Herodotus to Caria, which was a semi-greek semi-thracian city in the middle east.