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From the pictures and the cover diagrams, it looks like you are right, in that the terminal board is not wired correctly as it is.
I'm not absolutely sure, but it may be that terminals U1 and U2 are merely points of convenience. They don't actually do anything except to make it easy to tie the line power wires into the terminal board.
Also, from the inside cover wiring diagram , it looks like the V2 terminal is not connected to anything.
From the cover diagram, the run capacitor is always connected to the same two terminals, for either motor direction. If that is correct, then go ahead and wire the black and white capacitor leads to terminals V1 and Z1.
The start capacitor should be in parallel with the run capacitor, but it would have its own switch .... in series with it.
The DPDT switch will have the configuration shown here, looking from the bottom connection points. Wire the line power wires directly into the center terminals of the DPDT switch:
Z2 φ V1
V1 φ Z2
where the left side is CW rotation, and right side is CCW rotation.
The first thing you need to do is to sort out the wires coming from the DPDT switch. Determine which two wires are the center terminals, which two are one side, and which two are the other. ...