Speaking only for myself I think of "air" as the space in between and around the instruments. Like the musicians are in a real space. Typically a "big" sound stage has lots of air, like the musicians are in a large hall.

I'll venture a response, though others may have a different take.

One of the intrinsic challenges in the language on audiophile fora (or whatever you care to call them) is that people are often communicating through either relatively concrete and well-defined engineering, electrical, and acoustic concepts (where there are standard definitions, metrics, and equations), or they are trying the altogether harder and decidedly more subjective task of translating highly personal auditory experiences into language that others can understand, generally through visual and physical metaphors.  This is reasonable since part of the allure of good audio is that it can recreate moments in time that are surprisingly real, concrete and visceral, even though it draws on just a subset of the senses.

In my experience, the use of "air" on this and other audio fora is emblematic of that.  While soundstage and depth both have the benefit of being more intuitively understandable, "air" is almost by its nature more abstract, and the divergent usage reflects that.

I've seen it routinely used in at least two distinct but interrelated ways, both here and elsewhere. (I did a quick review of the non-literal references to "air" on the Decware forum to confirm this impression, but please don't ask me for links. It's too late for footnotes for me.)  

In one sense, I see air referenced in close association with space and soundstage.  This gets to your point about resolution.  Where soundstage might refer to the distance between performers or the overall sense of where action is occurring in space and time on the recording, "air" is perhaps more intimate. It is the space and time immediately around a performer--the fine detail; and higher levels of resolution. Steve referred in one post (maybe on the 300B thread, though I can't be sure) to watching the arc of a drumstick between strikes on a triangle. My own sense is that is a combination of time and "air". That feeling not only of the physical distance between players, but the distance around them and how they are moving through it. It's the staggering sense that, because of the subtle auditory queues in the recording, I can not only see where Shirley Bassey is on the stage, but how she shifts her stance and the orientation to the microphone from one phrase to the next. Or, to bring it back to Steve's percussion analogy, that feeling of watching the cymbal strike shimmer and decay above the auditory register, rather have it stop with just the shimmer. Is it subjective? No doubt. Does it feel real in the moment? Incredibly so.

The second, distinct, and yet related sense in which I see air used in this forum and others is more particularly with respect to higher frequencies, where the "air" is the room around and above the highest audible notes. Perhaps, as an attempt to describe the space above the audible that the sound keeps moving in to.  Given that the higher order harmonics that give many instruments their individual character continue well into this range, above the range of human hearing but not necessarily of sensation, my own admittedly amateur view is that this second sense is closely related to the first--as reflected in the cymbal example above.

Again, folks are trying to communicate how one form of sensory perception (hearing) can profoundly recreate others (sight and touch). Some fuzziness and uncertainty in the translation is pretty reasonable.

Still, and as always, YMMV. It may simply be that my limited understanding of "air" is evidence that my capacitors have too few (or perhaps too many) microfarads, or that my speaker wire is not sufficiently isolated from the floor.  We live and we learn.

In my experience there a few different kinds of air with the hot kind predominant.

All good stuff here as well.  Here's my take FWIW:

Starting with the proverbial 3 blind men/persons describing an elephant.  Each man/person touches a specific area of the body of the elephant. One says an elephant is like rope (tail) another elephant is like a wall (body) finally the last says an elephant is like a hose (trunk).  All are correct but the context eludes them and so the description and understanding is incomplete.

That to me is the challenge of defining the AIR/AIRY as used in audiophile terms.

To me AIR is the combination of all of what has been already stated:
Separation, space, detail, decay, etc.

in contrast a lack of AIR is congestion and a blurred presentation because any one of the elements is missing or not as resolved as it could be.

By the way, I'm not intimating that any of the previous posts are the result of blindness...JIC.  It's all good!

This is some fantastic feedback!

Joman: Love your analogy.  Your conclusion is what I was also getting at—that it’s just not one thing, but a combination of factors.

Same Old DD: Glad to know I wasn’t the only one in the room struggling with the concept. Even though your question was rhetorical, I just wanted to clarify that my example of strings, was really not my example, rather another persons take on it and I was actually questioning, why Air should only be associated with higher frequencies. You definitely have a good answer there—that Air doesn’t necessarily reside only in the upper registers, but it’s easier to discern, when it is.  It’s funny, I almost mentioned the same thing—that you know it when you hear it.