Pic 2 of your first post + manual pic. It is a self adjusting system like you are thinking. The teeth would grab the shaft in one direction but not the other. This would let the shaft continue to self adjust in the direction of wear but not the other way around. Free play is controlled by the groove on the cover side of piston. This 1.8 to 2 mm spec isn't something you can change, it is more of a worn part indication.

There is a snap clip 40a that you remove so the pin 37 c can come out of the cover. This way you will be able to assemble the toothed washer ( curved portion up ) from the cover side of the pin , install snap ring into the piston then put the cover on and press hard enough that the snap clip engages.

The end goal is to have the piston and cover close together after assembly. ( Per manual , don't pull it apart otherwise you will have to start all over again. ) Once the trans shifts, the piston will move away from the cover, self adjust but not fully return