I've had 2 different sets of counter stools and they both swiveled. The first set when my kids were toddlers had backs. The first set went with us to a house that had only an island for dining in the kitchen (the kids were 5 and 8) and they did great with them.
The second set I owned also swiveled, had low backs and hydraulic lifts. These went with me in 2 different homes and were used at counter height and then bar height.
So my vote, is for 3 elements: swivel, back, and hydraulic/adjustable height! And I agree with other posters on the swivel that goes back to the original position.
Q