I love Paul Foxton, who made that video you embedded.
Here are the basic principles that I follow for mixing lights and shadows, in order of preference:
1) Add a lighter or darker color of the same hue. E.G. use a darker yellow (yellow ochre) to make a yellow shadow.
2) Add a lighter or darker color(s) of a nearby hue. By that I mean nearby on the color wheel. E.g. orange is near red, purple is near blue, etc. And use a good color wheel, if your color wheel doesn't have cyan and magenta, it is wrong.
3) Add a complimentary color or black, but re-saturate the shadow with a brighter color of the same or nearby hue. The bright color must be brighter than your original base color, therefore you should not use your brightest color as your base color.
Let me take you through an example. Let's say we are painting a tan object, such as a khaki military uniform. Khaki is orange in hue, with a very low chroma. So to mix a shadow color, we start with option 1, a darker color of the same hue. So we need to add a dark orange, with low chroma. So we use a brown, such as burnt sienna. Everyone knows brown is orange in hue, right? So we are using a dark orange (brown) to make a shadow for light orange (khaki). To make a "highlight" you'll have to add white, but the resulting color will probably be too low in chroma. So we can add chroma by a tiny amount of a bright orange. And when I say tiny amount, I mean tiny.
Some versions of khaki are yellow, not orange. So what do we do then? First choice would be a dark yellow (aka olive drab) for the shadow. If you don't have a dark yellow, you can move to option 2, and use a nearby hue, such as orange. Dark orange is brown, so you can use a yellowish (not reddish) brown such as burnt umber. Again, mix white for your "highlight" but re-saturate with a small amount of bright yellow, such as cadmium yellow or hansa yellow.
This formula works for any color.
