This is what I have found when using swords (1x of which was taken to approximately 30,000 grit waterstones.)
While I am terrible at sharpening, I do understand the effects of sharpening having ran through a few dozen packs of sand paper for polishing of blades.
Anything between 80-320 grit will leave a very tooth edge and will cut just about most materials but will do poorly for push cutting until about 320 grit (when you can finally get shaving sharpness and push paper cutting if your really good at it).
There is a downside to using this level of grit, you can easily get burrs, and uneven toothy edges due to the amount of material removed with each "strop".
My theory is that as the edge is being polished by the 600-1000 grit paper, the edge's burring and toothiness is removed however something happens at a very moment as well. While no where near the level of 320 or lower grit, 1000 grit still removes quite a bit of metal so as you remove toothy edges you leave "flatter" areas as well as leaves tiny scratches in the blade following to the end of the edge resulting in little crevices that result in areas with larger surface then not allowing for your thumb to drag without being cut, but having an angle those edges that is shaving sharp.
As you move to 3000 grit and higher, the scratch marks get smaller and more dense, and as you being to polish out the edge, the crevices get smaller thus the areas with "more" surface diminish, and you bring back the drag of your finger. But it isn't the same drag as the toothy edge, this time it's just your skin being cut open, unlike the toothy edge (if you wiggle them just so slightly, you will NOT be cut).
I think that's what Murray does with his three fingers test, once the edge is pretty much flawless you should be able to just slightly wiggle, and feel the blade sink in, and eventually stop you from wiggling.