Doug, your test displayed in Image #5 shows a clear difference in sharpness when the 2X Mark III extender is stopped down (from f/8 to f/11). But this seems to be a general characteristic of extenders: they give better results when stopping down. This is certainly true of the 1.4X Mark II when used with the 500mm f/4 lens, and I'll bet it's true with the 2X Mark II as well. In other words, without showing how the older 2X behaves with the 500 mm f/4 lens when stopped down, I don't see your result in Image #5 as necessarily showing the new 2X to be superior.
Just to put more fuel on the fire, in the direct comparison between the old and new 2X (Image #4, both taken at f/8), there is a barely detectable difference in sharpness, which is probably easily explained by the tolerance of the AF system.