For the examples, lowest voice is always C.

Interval construction triad (suggested) 7th e.g.
P5,P5 4-3 M suspension?   G (4-3 sus)
P5,m6 m(root)   Cm
P5,M6 M(root)   C
m6,P5 M(1st inv)   Ab
m6,m6 +(symmetric)   C/E/Ab aug
m6,M6 m(2nd inv)   Fm
M6,P5 m(1st inv)   Am
M6,m6 M(2nd inv)   F
M6,M6 o(2nd inv) 7(3rd inv) D7

The fact that all the basic chords appear on here means we can construct a very sturdy (if still incomplete) tonal system with just those three intervals.

The augmented chord suggests F, A, C#, melodic minors.