What is the nicest STOOL for a MODERN KITCHEN  

Depends...
On what look you are going for... But I would certainly choose the LEM over Stool One for contemporary modern.

I like the Covey stool (Jeff...
I like the Covey stool (Jeff Covey for Herman Miller).

i just bought at DWR 2 ...
i just bought at DWR 2 bertoia wire counter chairs made by knoll, they are really comfortable, i sat on a fake one at a bar two weeks ago and I liked to die it was so awful uncomfortable . I can honestly say fakes in this case are no good!!!!

My Vote
While not at all in the style that your looking for, my vote goes to Cherner.
http://www.chernerchair.com/index.php

It depends on the...
It depends on the kitchen....but I like Aalto and Buck

I like
Onda, Jamaica, Perriand Les Arcs, Mies Four Seasons.

All great so far !
Also love Bassam Fellows (wood) +
J. Morrisson's HiPad for Cappellini (fabric or leather)

+
LaPalma Cubo

Morrison high pad
.

If you're looking for somethi...
If you're looking for something with a sense of humor, there's always Castiglioni's Sella bicycle stool

Back support
I like the look of several stool designs without a back, but I find that with no back support, the amount of time that I can sit in one is very limited. That is, when I'm somewhere there are stools I'm forced to use. My home does not have an island in the kitchen, a breakfast bar, or anywhere else that would call for a high stool, and for some reason I'm very grateful. High stools just seem to be my least favorite seating solution. I just can't help but feel like if you want to sit, sit. If you want to stand, stand.

C&B "Low Back" is Somewhat Comfortable
I have a set of the HM Covey stools, and I cannot sit on one through a full meal - they look great but are not very comfortable. I replaced the Coveys with a bar height low back stool from Crate and Barrel named "Felix". I can make it through a meal and more sitting on one of these. Frames are very solid and the bent plywood seat has a slight amount of give to it. I went with white to match our Nelson Swag Leg Armchair used with an adjacent desk.

Heh-heh. . .
It's a little hard to understand why they photographed a white object on a white background. . . isn't it ?
Did George Nakashima ever do a (high) stool ?

Yes
He did....