Are you sure you're going all the way off and on and not just toggling between on and accessory? Just trying to come up with an explanation. I just went out to my '06 TJ and if I turn the key on, wait till the odometer comes on, then push the button above the ignition and turn the key all the way off, then on, all the way off, on, off and on again, it displays "done" on the odometer (since I'm not pulling a CEL at the moment). Also, if I hold in the trip button and then turn the ignition off it says "06-02", whatever that means.
Edit: In your video you're not turning the ignition on first, maybe that's your problem. I have to turn it on, wait for the odometer to light up, then go off on 3 times to get it to work.