Electrical – Does this speed controller alter resistance or capacitance

I was recently (today) helping someone with their home renovations, including replacement of the three-speed fan controllers. The old one, which we ripped out and replaced, is shown below. Note the crude circuit diagram on the black box object (the fan speed controller.)

I infer that the fan controller works by inserting a capacitance into the fan's power supply circuit. The slow speed is obtained by using the 4.3uF capacitor (purple, P), the medium speed is obtained by using the 2.1uF capacitor (red, R), and the high speed is obtained by direct connection (no capacitor.)

Here's a circuit diagram for a very similar three-speed motor controller by Clipsal.

I don't understand single-phase induction motors well enough to state with certainly exactly how the extra capacitance modifies the speed of the motor. Sorry!

Note that single-phase induction motors aren't self-starting and must include a phase shifted auxiliary winding to provide starting torque. The phase shift is provided by a capacitor, which may be relevant. (If anyone knows more about the workings of single-phase induction motors, please come forward!)

The circuit I believe you're looking for is something like this:
(please excuse the odd symbols for the switch & motor ...)

^{simulate this circuit – Schematic created using CircuitLab}

With the switch in the Low seting, only C1 is connected in the circuit.
When the switch is set to Medium or High, either C2 or C3 will be connected in parallel with C1, giving you a larger capacitance which (I believe ...) will produce a faster fan speed.
It would not be uncommon for the capacitance value for the Low speed setting to be low enough that the motor won't start spinning by itself.
Ceiling fan speed control switches are usually wired such that the switching sequence runs Off - High - medium - Low - Off, so that the fan starts up with the full-speed capacitance to get it going.
You'll probably need to do a bit of experimenting with capacitor values, but a rough guess would be to choose a full-speed total capacitance of about 10x the value which just barely allows it to start.