As far as I know the Hawaiian and Scandinavian derivations given on this site are the only correct ones for
Kai as a name. Many of the other "meanings" may well be correct for the word "kai" in various languages. It is always good to know what meanings names have as words in other languages, but that doesn't mean they have been used as names in those languages or that those meanings have any relevance for the historical use of the name.
The Swedish name dictionary I own (
Svenska förnamn by
Roland Otterbjörk) says that
Kaja is a pet form of
Karin and does not list the spelling
Kaia, and the Norwegian name dictionary
Norsk Personnamnleksikon basically gives the same derivation, deriving
Kaia from
Katrine. I've often wondered why Scandinavian experts don't also see
Kaia as possibly being a feminine form of
Kai, but all the ones I've seen think of
Kaia as a pet form of
Katherine and don't relate it to
Kai.