"Česky" or "Čeština"?
Well, currently the list is very simple - it has only one name for every language and its most visible placement is the interlanguage links list, so you should probably decide according to that.
Well, as Michawiki said, „česky“ is an adverb, while „čeština“ is the name of the language. As there is no simple definition of the contents of the $coreLanguageNames list, we have to decide what should be used in the only use case, i.e. the interlanguage list.
I would say both of the variants could be used there, and looking to the other Slavic languages, I see e.g. “Slovenčina”, which is an equivalent of “Čeština”, but e.g. “Polski” is (IIANM) equivalent of “Česky”.
I prefer the adverb for interwiki links because I would understand the interwiki link text as "in Czech", "in Upper Sorbian", "in Polish" etc. Behind the interwiki link is no page about a language but the article in the corresponding language. But I think the Polish form Polski is an adjective, the adverb should be po polsku.
FYI interlanguage list is not the only use case. It's also used in language selector in preferences and other places.
I think most of such “generic” uses (just a simple selector with no surrounding sentence or other such context) could use both adverb or noun. The problem would be in cases somebody would like to include the language designator into a sentence (“If you want to read this in $1, click $2.”), but that cannot work at all in flective languages like Czech, anyway (you would need to decline the word).
So – I agree with Michawiki that keeping “česky” to be used in interlanguage links and language selectors is better. Even though it is not exactly a “language name”. (Note this also explains the difference between MW and CLDR.)