You can't controll a motors speed without either a) having the motor speed coupled to AC supply frequency or b) changing the voltage (works for AC and DC, depending on design).
The early washers used method a). Their motors had several sets of windings for different speeds, reversing polarity reversed the direction. Frequency was set by supply from the socket. For example a lot of dryers use that system as they only need 1 speed for the most parts, and barely ever need to reverse.
Modern motors that have a variable speed controll and use either method a) by the means of an inverter that can put out AC at variabe frequencys or method b) via a voltage divider of some form.
The wiring diagramm for the GE shows that the machine with that model number is a single speed machine, so it uses the fixed frequency of the wall supply to determine its motor speed. (See screenshot I attached). Clutch/Transmission are used for shifting from agitation to spin.
The second pic just shows a circuit. That circuit can verry well be part of the MCU (Motor Control Unit) or conected to a MCU. The main PCB can communicate with the MCU via a protocoll simmilar to the USB protocolls used, and USB 2.0 only uses 4 wires: Power +/- and data +/-. GE did this on their Hydrowave washers, which, if you search the trouble shooting videos for, have a board mounted on top of the motor, with a indicator LED. That board checks the mode shifter position and controlls motor speed.
So, to achieve multiple speeds with a PSC motor, you have to go the same direction as any other mutiple speed washers: Either MCU or different windinings for different speeds.
