Depending on the inversion, the chord containing the notes C Eb G A could be named A half-diminished or Cm6.
How do the two chords differ harmonically? Is there an example of a chord progression where the harmonic analysis has to use use one of the chord names because using the other name would make no sense? Or are the chord names synonymous?
The two chords differ in the way they are used in a harmonic context. A half-diminished chord is unstable because of the flat fifth (from the root), whereas a minor 6 chord is stable (it contains a perfect fifth from the root).
A simple (and very common) example would be a II-V-I progression in a minor key. E.g., in the key of A minor you could have the following II-V-I progression:
Bm7(b5) - E7 - Am6
Here, the Bm7(b5) would usually be interpreted as an (unstable) half-diminished chord, leading to the V chord, which in turn leads to the I chord. The harmonic context strongly suggests the Am6 chord as a stable (I) chord, and hence it is interpreted as a minor 6 chord, not as an inversion of a half-diminished chord.
I'm convinced that it's not the inversion but the harmonic context which determines the quality of the chord (half-diminished or minor 6). In the above II-V-I progression I hear the last chord as a minor 6 chord, whether or not the note A is chosen as the bass note. For my ears it remains a minor 6 chord even if the F# is the lowest note. In the latter case, the chord would sound like a half-diminished chord if played in isolation without any harmonic context.
There are cases where the chord quality can remain ambiguous, such as the first chord in the above example. It could also be interpreted as a Dm6 chord, making the progression a IV-V-I progression in minor. Ambiguous cases occur if the harmonic context does not strongly favor one or the other interpretation. In such cases it is usually the inversion that suggests the interpretation as a half-diminished chord or as a minor 6 chord. But these cases are rare, and in many practical situations the harmonic context clearly defines the quality of the chord.
