While I was reading about commonalities between French A6 and tritone substitute, I kept bumping into the similarities between French A6 and V7b5/b5.
French A6 is ^b6 ^1 ^2 ^#4 and V7b5/b5 is ^5 ^7 ^b2 ^4.
Their scale degrees and interval structure are different. But what similarities are there to draw between them?
Note the passage's wording carefully. The French Augmented 6th in question spans ^♭2 to ^7 instead of the usual ^♭6 to ^♯4. This means that the French Augmented 6th in question contains ^♭2 - ^4 - ^5 - ^7. These are the same notes as in V7♭5 in second inversion.
You can look at it two ways. Let's take an example in C minor where the French augmented sixth chord will be spelled A♭ C D F♯:
A♭ C D F♯ can be enharmonically respelled to A♭ C E𝄫 G♭ which is A♭7♭5.A♭ C D F♯ can be viewed as an altered iio7 chord, which would be spelled D F A♭ C, where the alteration is raising the chord's third to create D F♯ A♭ C, which can then be put in second inversion for the common Fr+6 voicing A♭ C D F♯, and also it can be viewed as yet another altered dominant, because D F♯ A♭ C is D7♭5.In the quote you posted, the concluding "more rarely..." part is new to me and interesting. I would recommend comparing those resolutions to the resolution of a common tone diminished seventh chord to a major tonic triad, where the common link between the various chords is the shared tritone of ^6 and ♯^2 or ^1 and ♯^4.
The two rare chords the author presents are analogous to a tritone substitution of a dominant chord, except the chords described are tritone substitutions of a common tone diminished seventh chord.
There is a lot of wacky enharmonic spelling changes, substitutions, and re-arrangement of pitches involved with these concepts. If something isn't clear I can add additional detail.
Of course the very, very important thing to distinguish is how these chord resolve. The Fr+6 standard resolution is to a dominant chord, and to be very standard it will be a triad with a half cadence feel. The point is they are subdominant or pre-dominant in function. Compare that with a jazz tritone substitution which is dominant in function, and the plagal resolution of the "more rarely" used chords.
