Thanks for the update, Raul. I'm very glad the governor slider had not come off the end of the shaft inside the crankcase. If it had, the governor would not work at all, and the engine would run at self-destruct speed any time the speed control was set at anything above idle.
I'm accustomed to pressure washers being run at full governed speed at all times, rather than having an operator-accessible engine speed control. On the washers I've seen there is a screwdriver adjustment of the full governed speed, but no need for the operator to interfere with it except during engine tune-ups: in service it always runs at full governed speed. I'm aware though that some of the popular units in the US do have an operator-accessible speed adjustment lever, though I'm not clear on why.
The system for controlling engine speed stretches the governor spring when you want a speed increase, and relaxes it for a decrease. The centrifugal governor inside the crankcase pushes the governor slider in the direction to close the throttle, with a force proportional to engine speed. The centrifugal governor and the governor spring thus oppose each other, and reach an equilibrium at some speed where the force from the centrifugal governor is exactly equal and opposite to the force from the governor spring.
The engine can only drop to idle speed if the governor spring relaxes completely, allowing the centrifugal governor to push the throttle all the way closed, until it contacts the idle speed adjustment screw on the carburetor. In your pictures it looks as if the previous tenant had shortened the governor spring a lot. In effect this would have set a minimum speed the engine can run at. With the correct, full length governor spring, it should be possible to fully relax the governor spring, allowing the centrifugal governor to push the throttle right back to idle position.
Note that the way I use a pressure pump, the engine speed is fixed and simply runs at maximum speed except when I stop the engine, which I routinely do whenever I won't be spraying for a minute or two. It seems to me that it is as easy to stop and start the engine as it is to change the engine speed setting down and then up again. Most of the larger, commercial pressure washers simply run all the time at full speed - their pumps are rated to tolerate the frequent dumps of hot water required to keep the pump from cavitating. Light duty pumps for home use treat the hot water dump more as an emergency protection than a normal part of operation.
I have never attempted to run a pressure pump with the engine idling. It is possible that the basic load imposed by the pump will be more than the engine can support at idle, so the engine would stall. That may be why the governor spring on your pump engine has been shortened.
