I had the 1,4x II that I replace with the III series. I confirm that the old one works better with the older lens.
The main difference between both is that the new one resolves a little more detail on the longer distance, but I feel that it vignettes more than the previous one.
I had the 2x III. I sold it because my copy did not work so well. It was acceptable at a very short distance but worst at long.
I also have the 1,7x that is in the middle of quality between the 1,4x and the 2x, but I don't use it often. For me, it is better to crop an image shot with a higher resolution camera like the D850 than take the same photo with the D5+1,7x.
On the 500 PF, the 1,4x III works quite well, the quality remains acceptable, but the autofocus slows down. When I go out with the PF, the multiplier remains at home.
I use them only when I am in a hide with the 600FL and the tripod.