I know nothing of the physics of this, but the dark inside rind artifact is the same distance from the edge as the bright outside ring. Are you saying that they are not generated by the same process with one being an exterior diffraction ring and the other a corresponding interior diffraction ring?
It's complicated. You can think of the image as though it is an underlying sharp (under sampled) "true" image that has been convolved with the Airy disk. But that isn't quite right. If that were the case, you could at least in principle deconvolve the Airy disk back out if the SNR were high enough and you had a perfect knowledge of the true Airy disk, ending up with a perfectly sharp (under sampled) image again. But you can't, because that isn't what has happened. What has happened is that the finite aperture has filtered out the higher frequencies required to make a perfectly sharp image. You can't deconvolve them back out because they were not recorded. You can in principle deconvolve atmospheric turbulence if the SNR is high enough and you have a good enough estimate of the point spread function. So when we sharpen we compensate for atmospheric turbulence, but we cannot really compensate for diffraction effects. Hence when we image stars and sharpen them, we get sharp images of Airy disks, not single pixels containing all the light. If you try to do that, all you can end up with is noise, since that is the only thing that has reached the sensor at those frequencies.
Diffraction effects are the result of high frequency filtering, and the ringing that results from not having the high frequencies required to make perfectly sharp edges (or points, in the case of the Airy disk), like this "image" of a perfectly sharp square wave (like a cross section of intensity on a line cutting through Mars, for example), that is formed by having successively higher frequencies added or removed:
You can see that near the sharp edge the intensity actually over shoots above that of the "true" image. And just inside that it undershoots. That's the bright and dark rings of the rind. Sometimes you will see further bright and dark fringes of decreasing amplitude as you move in, but they are usually lost in the features and non-uniformity of the planet's intensity.
It's a mistake to blame this all on over-sharpening. It's a real optical effect that may be harder to see under some circumstances. It's harder to see in Jupiter than Mars because Jupiter has a softer edge. You don't see this effect on the terminator of Mars because that is essentially a very soft edge. When I say "soft edge" I mean that there are fewer, if any, higher frequencies required to make the image of that edge, and hence the high frequency filtering has less, if any, effect there.
This example is not exact, by the way; this is a truncated Fourier series on a finite domain. The image plane is not constrained in the same way. In this example adding higher frequencies brings the image closer to the true diameter of the true image. But in the real world the ringing is not constrained in this way, and the last bright fringe really does more or less correspond to the position of the true limb of the planet.
There is a faint ring outside of the planet (perhaps even two or three) that "corresponds" to the diffraction rings in a more or less straightforward way.