I’m not sure if this will help, but for what it’s worth here are my inexpert thoughts…
I think that the third picture is the most helpful to understand how the mechanism works and therefore how you might build or adapt similar.
It looks like the joinery is pretty simple on the leg frames (mortice and tenon or but joint – this one is probably screwed). The long back rail looks to be set in a housing but is probably screwed too for strength and construction speed.
Looks like there is a gas extension strut/spring on each side that helps to power the desk top lift. They are commonly used for divan beds and boot/tailgate opening in cars. These can be bought online and sometime in car/auto parts stores. The difficulty would be choosing one with the right size and level of “spring” to help lift whatever weight is to be held above
The holes in the extension legs line up with the hole in the top rail of the outside leg frame and it looks as though an appropriately sized dowel or peg is put through both to lock in whatever height is chosen. It’s not clear if there is any complication around ensuring that both sides of the table can be pegged in place at the same time or automatically locked/pegged at each detente as it rises.
The rest just looks like various tracks (think drawer runners etc) to ensure that the two legs frames can slide smoothly and within limits.
In my opinion, the design pictured doesn’t look very sturdy and I would want to make it a bit more substantial stock if I were making it. Possibly more cross bracing, too.