I believe the Nortons use a different rating system so in that case you are correct (Yea this makes things confusing). Good point. It's been a point of amusement when people call a stone a 'fast' fine stone only later to find that the stone actually is a coarser stone. Additionally even two stones rated with the same micron particle size often don't have the correct median particle size and can have differing 'distributions around that particle size.
I wrote this up a while back here:http://precisesharpening.blogspot.com/2 ... arter.html
andhttp://precisesharpening.blogspot.com/2 ... arter.html
andhttp://precisesharpening.blogspot.com/2 ... nning.html
So back to the original point. For another same grit stone using the same grit rating system, I think you would be better served by just going on to a finer stone for further refinement or using the one stone at say 8k a bit longer.
For reasons of personal sanity, I tend to use a conversion of grits to microns of 1k = 16 microns and 16k = 1 micron and deriving the rest from there, e.g. 4k = 4 microns 8k= 2 microns etc. Not perfect and could be criticized, but works for me.