I think King's motivation for his C1 and C2 distinction is good. I'm motivated to accept something like what he wants, though I would want to avoid his problems. I think the classical empirical-foundationalists also want something like what he has.
The motivation behind C1, rationally reconstructed, seems to be one's being able to get into the space of reasons, make material inferences, paraphrase, justify, etc. one's use of language.
The motivation behind C2, rationally reconstructed, seems to be one's being able to reliably report, assert, identify, etc. One has to be able to use the words correctly.
Someone like CI Lewis is going to say that C1 is basically being able to know what is logically implied by a term; and, it seems, what is analytically implied. Lewis thinks these are distinct, but King could be said to just conjoin a term's 'connotation' and 'signification' as Lewis uses the words (roughly).
Lewis would say that C2 is one's being able to recognize instances. One has to know what the term picks out, and all consistently thinkable cases where the term would pick out those things. Again, King could be said to just have conjoined the ideas of 'denotation' and 'comprehension' as Lewis uses the words (roughly).
Maybe Russell-Mill would take C1 to be one's knowing connotations, C2 knowing denotations.
These people all want to cash out 'word-meaning' and 'sense-meanings'. It seems that something like this distinction between 'inter-linguistic transitions' and 'language-entry transitions' is good to have. Even for Quine we need stimulus-meanings 'in presence' and 'in absence'. HH Price likes this notion of 'in presence' and 'in absence' so he would want C1 to be something like 'thinking of a term in absence' and 'thinking of a term in presence'.
So even if we don't like King's formulation, shouldn't we look to keep a sort of C1 and C2 distinction for linguistic competency?