Unless the song is some kind of esoteric or experimental song, I seriously doubt the composer would suggest to give you the choice to play either a C major or an A minor (Am) chord at the same point, because depending on which chord you play (C or Am), you would drastically alter the mood of the song at that point, because the C major chord would give a brighter, happier sound, whereas the A minor chord would give a more somber or sad sound.
I have seen music where a chord is stated, but then a simpler chord is suggested for people who aren't familiar enough to play the original chord. For example, a song might state to play a G9, but then also notate that a G7 could be played instead of the G9 for people not familiar with a 9th chord. But C major and A minor are both very basic chords, and again, they convey very different moods, so I doubt that is the case here.
I agree that in my experience, when a chord is written as "x/y", the "x" is the chord you're playing, and the "y" is the bass note. If it really is written as "C/Am", which is showing as a whole chord after the slash ( "/" ), then I'm really not sure what the composer has in mind, and I can only agree with the possible suggestions above. If it is written the wrong way around, then that would mean to play a C bass with an Am chord -- which, by the way, is the same as a C6 chord. Correspondingly, playing an Em chord with a G bass is the same as a G6 chord.
If the notation really does mean to play the two chords together (in other words, play a C major chord and an A minor chord together), then that is also very strange, because as noted above, when you do that, you just get an A minor 7 (Am7) chord. C major is C, E, and G, and A minor is A, C, and E. When you account for the repeated notes, you're left with A, C, E, and G -- which is exactly an Am7 -- or, if the C is the lowest note (C, E, G, A), then it is a C6. I would think the composer would just simply write the intended chord as Am7 or C6, rather than some confusing instruction to play two different chords at the same time. By the same token, playing G major and E minor together, and then accounting for the repeated notes gets you E, G, B, D, which is Em7, or G, B, D, E, which is G6.