Electrical – the device resembling a capacitor in the ceiling fan

You are not wired the same way as the picture in your instructions. I think the power comes in to the ceiling box, and the switch is a separate cable.

To verify this, disconnect all of the wiring in the ceiling box and at the switch. Turn the breaker on and carefully check for voltage between the black and white wires. When you find the black and white pair (or just the black wire if you are using a voltage tester) that has power, then you can turn the breaker off.

If this test confirms voltage on just one black wire, then connect as follows:

1. In the ceiling box:
   • Connect the black powered wire to the other black wire in the ceiling box.
   • Identify the white wire that is paired with the black powered wire. It will be in the same cable sheath. Connect that white wire to the white wire on the fan.
   • Connect the remaining white wire (which would be coming from the switch) to the black wire on the fan/controller. Wrap a piece of electrical tape around the white wire about a half inch back from the wirenut. This will signify that the white wire is carrying live current and is not a neutral.
2. In the switch box:
   • Connect the black wire to the switch wire with the yellow sticker.
   • Connect the white wire to the switch wire with the blue sticker. Also mark this white wire with electrical tape.

I came up with a solution so I'm going to to go ahead and post it in case anyone else should happen upon this and find it useful.

I added the 1-in, 2-out, 4-position speed switch into the diagram. The switch works by taking one input and routing it thusly:
[1: (off) | 2:A | 3:B | 4:A+B]
and in this way controls speed by energizing either one part, 2 parts, or all 3 parts of the run (inner) winding (see diagram to help make sense of this).

I found conflicting information about achieving reverse (some sources said switch the polarity of the run winding, some said the start winding); I went with the start winding (the outer one) and it worked. Lastly, I found the capacitor was to be inserted in the start winding and that was straightforward enough.

With the AC input I made red Hot and blue Neutral; however it doesn't appear to me it would make any difference, other than that one way the reverse switch is up for clockwise rotation and the other it is up for counterclockwise rotation.