My chair suddenly started lurching precariously from side to side, and when I took a closer look I noticed this inner ring around the cylinder that keeps the chair centered had popped out and vomited a bunch of grease everywhere.
I can push this thing back in and it'll sort of click into place but a few minutes of sitting later the chair starts wobbling again.
They do sell the tool-kit for removing the pistons by itself on the aforementioned website and probably other places. It's basically two ring collars that snap/screw together with a screw and hex key to turn it to push the pieces apart though that may be the easy part as the pistons tend to be pressed into the base and getting that out can be a trouble and require some bashing with the thing braced and in some cases just may not come out without using a method that destroys the piston (not a problem when replacing it but a big one when trying to swap).
The one in your picture may be a cheaper design than what I'm thinking of here. I'm not a chair expert I just once replaced a cylinder on an office chair I salvaged and followed the instructions and watched a couple videos.