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I’ve seen a similar demo with just the toothpaste. Once I seem to remember someone using melted chocolate getting pretty impressive results.
Unless they’re using “white chocolate” or carob, that’s unforgivable heresy. I might forgive a low-cocoa milk chocolate, if done For Science™, but otherwise… 
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I guess pretty much any smooth paste that can hold some mild aggregate, such as the baking soda, will work?
Forget it. Headlights are not “faded” they have a UV coating that has failed. In some cases it’s the outer skin of the acrylic that’s crazed.
You can use pretty much anything you like to “polish” the outer surface, but products meant for the job (like wet and dry paper) will do it better and more efficiently, and it won’t last long if you don’t give it a dash of UV protectant (clear paint) when you are finished.
I used to sand down through the grits to 2000 grit then polish with a rubbing compound for a perfect finish, but it’s much easier to start at about 600, 800 then 1200 then squirt with a couple of coats of spray lacquer which turns everything a nice shiny clear (if you are confident with the paint you can just stop after 600 and go for it. If you can buy cans of 2k automotive stuff economically enough, the job will pretty much last for the rest of the life of the car.
It’s best just to use a protective “polish” like Meguires plastx or clear headlight coating, then you won’t need to do any of that!!
I know about this stuff because, apart from being a bit humid where I live, the AVERAGE UV index is somewhere between 12 and 14 so I’ve had a bit of practice cleaning up headlights.
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