So.... what is it exactly that we are looking for in all of these anyway 
Each one looks a bit different from the others but what exactly makes for a better or worse downsize?
I think there are two features that are appearing again and again in some resizes that I find undesirable:
1) A solid, clearly delineated grey area in the middle,
2) False patterns outside the Nyquist circle (in your factor 8 resize it would be the eighth circle).
The first feature suggests that the low-pass filter (blurring) applied before downsizing is too strong and has a relatively hard cutoff. Therefore we see detail up to a certain point and no detail (also no false detail) beyond that. A real image will likely have some smearing effects, similar as to when too much noise reduction is applied. But this may be tested on an actual image to see whether this intuition is correct. It might look ok with images that are more graphic in nature (no fine textures or none that are interesting), less ok with natural objects where we expect fractal detail (e.g. we are happier to see something that resembles grass, even when the grass is too thin to be resolved, instead of just a green smear. Because the grass is irregular, we are unlikely to see the interference that plagues the Siemens star).
The second feature is due to oversharpening in the detail that can in principle be resolved accurately. In regularly patterned subjects, this will lead to funny patterns that are clearly recognized by the viewer to be fake.
The LR resize and the 'Save for Web' resize of PS don't show either of those two features. The small detail smoothly fades away towards the center and outside circle #8 there are no (or very weak) interference patterns.
