I did a search for the minor 6 chord and I found that this refers to the minor triad with an added major 6th interval (or 13th rather). I am studying a song that has a minor triad with a minor 13th interval. What do you call this chord?
It's quite happy being a m6, with the major 6th note just above the P5 - as in A C E F#. It doesn't have to be 13, in fact it usually isn't. Flatten the major 6 to an Fn, and the notes are A C E F. It's not an A minor derivative now, but an Fmaj7. With the A as its lowest note, it's known as Fmaj7/A. The first inversion of Fmaj7.
The chord symbol for a minor triad with an added minor 6th would be Xm(♭6), where X is the chord root.
This requires that X is truly the root of the chord. Otherwise, as Tim explained, it's an inversion of a maj7 chord.
For the chord to be considered a 13th chord, it would be implied that the chordal 7th is present. [C E♭ G A♭] would be Cm(♭6); whereas, [C E♭ G B♭ A♭] would be Cm7(♭13).
The main distinction between a m(♭6) chord and an inverted maj7 chord is the point of stability. Given [C E♭ G A♭], if the C is the stable pitch, then the chord is Cm(♭6); if the A♭ is stable, then it's A♭maj7/C.
Domenico Scarlatti uses m(♭6) chords in his Sonata in C Major, K502.
measures 82–84: See beat 3 of measures 82 and 84 [A C E F]
measure 88: See beat 3 [G B♭ D E♭]
(IMAGE SOURCE: IMSLP)