I'm all about some healthy chord tone discussion... but we always need to take into account that these things can be somewhat regional, colloquial, and dependent on whatever scene or subculture you're in. These things haven't been codified to the extent of the common practice music of a European conservatory.
The Aebersold nomenclature works well for me, probably largely because one of my beloved teachers, Dan Haerle, helped to develop it (or at least I heavily suspect he did, he and Aebersold were lifelong friends). Extensions can certainly exist in qualities other than dominant, although the way they behave in practice usually requires some alteration... Most commonly, in any chord with a major 3, we usually like to raise the 11 by a half-step, except for sus4 candidates. With a minor chord, we usually like a major 6 or 13 instead of what we would expect, a minor 6 or 13. So our "default" fully extended chords would sound something like this:
Major 13: 1, maj3, 5, maj7, 9, #11, maj13 (corresponds to the Lydian sound)
Minor 13: 1, min3, 5, min7, 9, 11, maj13 (corresponds to the Dorian sound)
Dominant 13: 1, maj3, 5, min7, 9, #11, maj13 (corresponds to the Lydian Dominant sound) [corrected, thanks cappers!]
There are, of course, countless exceptions, alterations, diversions, and developments from these, but I find this framework to be a useful place to start in jazz. The "b13" in question here is actually usually a "b9" in practice, since we'd probably be referring to the pitch in relation to the chord at hand, and in bebop we frequently see this kind of movement on a V7(b9) chord. And not just bebop, of course, we hear it in minor-key western music going back to Bach and earlier.
In terms of improvising and composing, the extensions can usually simply be thought of as their corresponding pitch in the lower octave (2, 4, 6), for practical purposes. The extension numbers are good to be aware of in terms of how these notes would be stacked by default in tertian chord voicing.
Apologies for the long-winded and only questionably useful post, but I could geek out about this stuff all day... loved it since I was a teenager, for some reason.