The second one is a fixed pivot with guides.
The third one is using a slider, fixed pivot and a circle.
The fourth one is a normal chair with the lever to pull when want to adjust the height of the chair. But by doing that, we have to bend down a bit to adjust it. So I'd suggest that we add a linkage (look at picture 5). Then when you can use your foot to adjust the height of the chair, and it can act as a foot rest.