My speed regulator is clearly labeled as being 9-12V AC. I have an official 12VAC transformer.
I was curious about the internals of this device, so I opened it up:
The circuit begins with 4 diodes in standard bridge rectifier configuration. This converts the AC input into a rippled DC voltage. The large capacitor (2200uF, 25V) then cleans up the ripples. There is also an LM317 adjustable voltage regulator configured to output various voltages depending on the dial setting.
This means that the input can be 9-12 voltes AC or DC because the rectifier simply passes DC current.