Music has so many different languages within it, it's very tricky working out what the questions actually are. In guitar conversation, I tend to assume we are not talking about music theory, traditional notation and Chord numbers, as it then gets confusing for somebody who doesn't speak that version of the language. In a fake book C6 is pretty common, and as a chord just adds the A, but the important thing is that practically every rock and roll and blue song adds an A rhythmically flipping the G to an A and back with your little finger, and by extension you can follow the bass players fingers and run the G up to the A, then the Bb then C and to the ninth, the D - IF, it's appropriate - so in fake book language, thats C, C6, C7 and C9 all covered.
I always think of the differences in the chords telling the story, so in that sequence, your ear hears the ascending melody of the changing notes and it has to go somewhere, so it's the next chord that might be the critical one - what comes next? an Eb chord, maybe, or maybe not.
C6 sounds like C6 in isolation, but contains the same notes as Am7 - so Am, is subtly different to C6, as it misses out one note, but that's really it. I always love the discussions on what exactly the Beatle's Chord is at the start of Hard Days Night. Loads of absolutely correct answers (that are different) but what makes it work is the whole thing - just a great combination of notes. If you play the chord with different fingers on different strings, or on the wrong instrument, it's just close but not close enough.
I-IV-V type chord descriptions don't work so well nowadays because they don't get taught in school, at least here, so this way of numbering chords confuses. Worse is that chords are always taught in piano terms, never guitar, so see C, and most kids fresh from school can prod C-E-G, but play G-C-E and they have no clue!
As I'm qualified as a teacher, I figured I would fill in some spare time with some teaching - bad move! It is so dreadful, it make me cringe, so I don't do that any more. I think the kids who learn on guitar make a much better sound because they don't need to know the physics, just the sound, so a beginner playing guitar can sound pretty good, but somebody at the same musical stage playing piano sounds totally awful, because the piano player with all first position root chords simply shunted up and down the keyboard is just not how the piano was meant to be played. Guitarist who learn the bar chord E Major and minor fingering and just go up and down the fretboard can play almost every chord and thus every song, but very few learn to play like that - thank goodness!