Kitchen chairs are way too low. In a pool room, you want to be able to look down at the table to see all the action clearly from a perspective higher than the table. I knew that stools were the way to go. My criteria when I started shopping for stools were sturdy, back preferred but not critical, absolutely no arms, and a southwestern style. Our room is tight and arms on a stool would just add another obstacle to get in the way of shooting.
Backs are okay because they'd just be against the wall.
One of the first things I found out is that there are three sizes, or heights, for stools. The two most commons heights are counter height and bar height, approximately 23-25 inches and 30 inches, respectively. An even taller stool than the bar stool is the spectator stool. I didn't see any of those, but I was told about them. I imagine they are around 35 inches tall.