More links where you can see that a Major 6 chord is made of (1,3,5,6)
D6=(D,F#,A,B)http://members.aol.com/chordmaps/part4.htm
C6=(C,E,G,A)http://www.guitarsite.com/chords3.htm#naming
C6=(C,E,G,A)http://www.jazzguitar.be/jazz_guitar_chord_theory_3.html
So, it is not correct to compare the two chords.
C6=...............(C,E,G,A)
Am/C=...........(A.C,E,)/C