The motor in the car is completely irrelevant. The only thing the servo controller sees is the PWM throttle signal from the receiver. I'm saying that to hook up the reverse lights, you do it exactly as you did with your forward lights, but you turn the LED around on the servo controller's output. You can run the forward and reverse LEDs off the same servo controller, you just need the reverse one wired backwards. The servo controller uses an H-Bridge to control the polarity of the voltage to the servo motor, so when you're turning one way, the servo motor gets a positive voltage. When you turn the other way, it gets a negative voltage. So.. if you parallel that servo controller to the ESC channel like you did in your video, when you go forward, that servo motor gets a positive voltage. When you go into reverse, that servo motor gets a negative voltage. So, if you replace that motor with a couple LEDs, one wired up "normal" just like in your video, and one wired "backwards", the "normal" one will come on when you go forward, just like in your video. The one wired backwards will come on when you go into reverse.

If I can find the time, I'll try to whip up a quick visio diagram and post it. It's super easy, but I'm sure my explanation doesn't make it seem that way.

Here's a quick diagram I drew up: