My point is that c chord progression that moves from A major to D major, then from G major to C major is in two different keys, and that a 6-2-5-1 progression, in the key of C would feature Am-Dm-G7-C.

If you think I am being nitpicky, play the two progression--even in a perpetual Foster's fog, you should be able to tell the difference. And don't ever use that "Folk Music pseudo Music Theory" again, or I'll clip you with me wobble board.