Oil paint is the most durable, better at hiding stains that may bleed through (just like what you have), and gives the best finish. The cons are that it will yellow a bit over time (but it will take a looong time). Is harder and messier to work with. You can't get it at lower or home depot, but an actual paint shop will have it. It has a stronger smell while it's curing too.
If you just can't do oil based, then water based cabinet paint is a close second.
I've used a bunch of different brands, and I can't say one is better than another so long as you buy cabinet paint. The cabinet in the video was Valspar cabinet paint (I think). The video washes out the color a bit, but it came out beautifully. If you use a paint conditioner, it helps even more - oil or water based.
The killz is probably a good call, but dark paints, and particularly oil paints are supposed to be able to hide. You can even get a hiding primer tinted the color that you are going with, so that may be as good or better than the stark while killz.
Good luck!