Your question and the linked video really seem like two different issues. You asked about voicings but the guitar lesson video issue is more about identifying a bona fide chord.
I think the guitar lesson video is transcribing the music incorrectly. The video's transcription - roughly tones
^5 b^6 ^5 ^1 b^7 ^1 over a complete, root position minor triad - sounds wrong when compared to the actual soundtrack.
From listening to the soundtrack, the first change sounds like root movement by third. That is a so-called weak progression with a characteristic feel. Importantly the voice leading uses an inversion on the first chord.
I think this transcription in
Cm is correct...
The chord symbols
Ab/C Cm Cm7 Cm seem OK. Notice the symbol is a first inversion
Ab major triad. It's a simple triad not a seventh chord. It isn't
Cmb6 (that's how the guitar lesson would label it) because there isn't a fifth (
G) present. That is what I think the guitar lesson video overlooks.
You might think:
Cmb6, six of one, half dozen of another. But that is being indifferent to inversions and what is a bona fide chord. It's bad harmony.
To me the essence of the harmony is
Ab/C Cm or in Roman numerals
Cm: VI6/3 i. It plays with the ambiguous feeling between major and minor. Importantly
C is the tonic and in this style either a plain
Cm triad or a
Cm7 could server as a tonic chord. Functionally it doesn't seem to make much difference and so the
Bb tone isn't very important in the functional sense.
Original answer about chord symbols, inversions, and enharmonic spellings:
My understanding is when jazz chord symbols are used the harmony is root position unless indicated otherwise with "slash" notation to denote what tone is in the bass.
With that understanding about inversions chords like
Am7 should not be considered the same thing. Both are assumed to be in root position so the former has a
C in the bass and the latter an
A in the bass.
If you indicate the inversion, then you could get genuinely redundant symbols like...
Regardless, the symbols don't tell you about closed or open voicing.
The only way I know to indicate specific voicing is with "drop" voicing. From a closed position chord various voices get dropped (inverted down an octave) and then you get a voicing with specific intervals. So a drop 2
G7 is voiced
D (P4) G (M3) B (d5) F.
I'm pretty sure I have seen lead sheets with a written note for a drop voicing.
...he's using (C E♭ G G♯) that he refers to it as Cm♭6?
It's hard to talk about this stuff when enharmonics are handled indifferently!
Cm♭6 would be
C Eb G Ab
Cm(add♯5) would be
C E♭ G G♯ the 'add' being necessary to indicate a perfect fifth and an augmented fifth are both present.
G♯M7 would be
G♯ B♯ D♯ Fx
Cm♭6/A♭ would be
Ab C Eb G but could be more simply labelled
AbM7 depending on what is the actual root and both are enharmonically equal to
G♯M7. Those symbols provide inversion info, so you know what tone is in the bass, but the symbols don't really tell the specific voicing (specific intervals between all the tones.)