I'm playing some chords from a song that goes Fm (F, A♭, C, F on guitar), Fmin7 (E♭, A♭, C, F), and a chord I don't know what to call that uses D, A♭, C, F. This appears to be similar to an Fmin/maj7 chord but with a major 6th instead of a 7th. Could this be called an Fmin/maj6, or would I need to call it something else?
I'm not quite sure why F-Ab-C-D would be considered closer to FmM7 than Fm7; in any event, your chord is an Fm6 chord. You could also interpret the chord D-F-Ab-C as a Dm7b5, or half-diminished chord.
The minor/major part of FmM7 indicates that the chord is an Fm chord with a major 7th, to distinguish it from an Fm7 with a minor 7th; note that the "minor" in Fm7 does not refer to the 7th, though, but to the minor quality of the Fm triad.
There is such a thing as an FmM13 chord, which would be spelled F-Ab-C-E-G-B-D, or possibly encountered as something like F-Ab-E-D.
If the lowest note of each chord is the note that changes each time so F->Eb->D, then the changing notes can be considered as passing tones, since they are not being used in a functional progression. For example, if the key center where this happens in F minor, then the Eb Ab C F chord could be changing the F minor to a ii chord in Eb. You don't mention what comes next, and that is the crucial chord to tell us if the Dm7-5 is functional or not. If the D moves down to Db or C, then we are still dealing with passing tones. If the D moves to a Bb7 chord, or a G7 chord, then it is functional.
So, if the D is part of the baseline, then it is not Fm6. Hope this helps!