There are several correct answers, but the best is clearly
C C/B Amin7.
Points other answer missed:
Second inversion chords are fairly unusual, and mostly used in cadences. They are particularly "weak", and there's no functional reason to call your second chord an Emin7/B. What is the chord doing? That's the entire question here and why we have the topic of "functional harmony".
"Em6 is actually confusing, as it's an Em chord, but the 6th bit is a major 6th - C#." No
"Both C - C/B - Am7 and Am7/C, Am7/B, Am7 seem correct in a chord progression." the key point is here progression... the first is a pretty normal progression, the second is not.
here's how you answer the question. Play the first chord. Do you hear that as solid and stable (root position major) or minor and unstable (first inversion)? Clearly, it's major root position. Then you have a dissonant bass note /B... simply a passing note and nothing happens harmonically until you settle on the logical Amin root position. All other things being equal, that's probably your best answer.
It is possible that rhythm and melodic emphasis could change this a bit. If the C/B were emphasized in some way then it might be heard as a Emin6, but I think that is pretty unlikely.
Think about what your chords are doing... where are they going... where are they leading? Realize that root movements of 4th and 5ths are the norm and are strong changes (2 of the 3 notes in the triad change)... movements of a 3rd or a 6th (notice these root movements are simply inversions) are less strong because the new triads have 2 notes in common... we dont hear as much of a change.... and root movement by step (or 7th but wtf?) are not usually perceived as progressions but more as tonal "shifts".
Hard to explain without sound, but I think your answer is there. C/B => Amin is a lovely dissonance and I think that's what you have here.
Spend time listening to chords and singing both the bass note and roots (and note they are not the same in inversions) until you start to get more of an intuition and feel for this. It takes time, but there's a whole landscape of sound out there most people miss.