(Courtesy of Frank Valdes, NSF's OIR Lab)
AVSKY is measured in each CCD after standard detrending and before any pupil and sky structure subtraction. This is done by the Automatic Cataloging Environment (ACE) system I developed. For the purposes of this discussion you can think of it as a SExtractor type of thing. ACE does a very careful measurement of the background in an image excluding sources and any masked data. One result of this is the scalar quantity AVSKY which is the average of this sky image that captures the background structure at a scale of roughly 200 pixels and is then used as part of the global pattern structure removal.
So, in the \(ooi\) data products (which I believe is what is in the Legacy Survey Data Release 2 catalog) each CCD has a different value of AVSKY and will vary due to gradients, the pupil reflections, and residual gain errors (corrected further later with a dark sky flat field). In the oki data products which apply a local sky subtraction (in a sense similar to what the Tractor does) the AVSKY value is the average of all the individual CCD values and is a global header parameter. The \(oki\) data also has the same effect of removing light around big/bright sources in a CCD while the ooi data by itself is less affected until sources are on the scale of a whole CCD or more.
The keyword SKYSUB is the average value of the sky across all CCDs after pupil pattern and large scale background surface fit to remove gradients and other non-local structure. The final sky that is derived and subtracted has the zeroth order term (the mean=SKYSUB) removed in order to leave the data with approximately the same count level.
I now realize that it is an oversight that there is no measure of the per CCD background after the pupil and low order background structure removal. In other words AVSKY should probably be corrected by the average of whatever correction is applied in each CCD but that is not currently the case. In as much as these large pattern subtraction steps bring the focal plane to a uniform background, SKYSUB is the best background value and AVSKY - SKYSUB would be a crude measure of the background pattern removed. Of course, whatever is done at the Tractor level with background would also not be represented in the AVSKY values.
I would also note that exposures taken with a bright moon, typically in the \(z\)-band, there can be a fairly strong background gradient and structure. Also the pupil pattern is significant in \(g\) and \(z\) and can reach amplitudes of several percent of sky.