"Česky" or "Čeština"?

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.)