Whether the three-way is a PAS or a throttle, the red is 5v, the yellow is ground and the blue is signal. If you look at the PCB, two blues join to the same pad (VL), the other two blues go to BK1 and BK2, which are the two brake signals.
It could be that the blue to the 4-way is sending on the throttle or PAS signal to whatever goes on the 4-way connector, but I can't think what that could be unless it's some sort of intelligent throttle that's only allowed when pedalling. What we know is that it has red battery voltage, blue signal wire common with PAS/throttle, black ground and a mysterious grey wire that goes to pad HV. Pad HV is used for the speed limit on the Cyclamatic version of that controller, but that just means it's an input to the CPU that could be programmed for anything.
The only thing I can suggest is to test it. Connect the battery, LCD and motor, then try a throttle on the three-way. See if it works. To be honest, I think the three-way is for a PAS, so if the throttle doesn't work try a PAS. You don't need to install the PAS. Just twizzle it around in your hand. Try both directions and the magnet disc both ways up. Don't forget that you need to set the LCD to at least level 1.
If it only works with the PAS on the the three-way, then the grey wire is probably a throttle signal, so try connecting a throttle with signal to grey, ground to black and get the 5v from one of the brake reds. see if that works.