Mooh: "A diminished chord on the 7th degree (in this case C#) often wants to resolve to the 1 chord (D). I'm sure you know this, but I'm struggling to understand something here. What am I missing?"

You are, of course, correct. I was trying, perhaps clumsily, to address the OP's observation that a C-E-G triad produces different chords depending on the key signature, which of course it doesn't, since in the examples s/he cited, it's no longer a C-E-G chord.