I read the question C major 6th chord and A minor 7th chord: same notes but different quality, why?, and it talks about the quality of the different chords. But when would you call a 7th chord first inversion a 6th chord? I'm sure there's other pairs of chords where this question could be applied also.
Context is always important when naming chords as many chords can be named multiple ways.
When given a choice between a chord like Am7/C or C6 the default is to call it a C6 for a good reason because have the bass note be perceived as the root is very typical rather than assume the chord is in first inversion.
In almost every case those two chords serve the exact same function, however it should be noted there is an obvious case where one would be preferred over the other and that is when the relationship between the chords of one makes more sense then the other.
The simplest explanation is in the case of when this chord is preceded by a dominant. If the chord before was and E7 or some variant of it, you would most likely want to the next chord as some kind of chord with the root of A to show the tonic-dominant relationship and in this case you would name it Am7/C. If the chord before was and G7 or some variant of it, you would most likely want to the next chord as some kind of chord with the root of C to show the tonic-dominant relationship and in this case you would name it C6.
If you're going to call it C6, it's because it's in a sequence where C is more predominant than A, for instance, in a sequence C, C6, Cma7, C6. If it was preceded by the dom.5 of A,(E), it would then take the mantle of Am7. If it was, say, C6, 1st inversion, thus, in isolation, could be either, then the chords either side would dictate technically what it's called, leaning towards the actual key - C or Am.