Yes, there are four methods that I know.
One, you're in a hurry, so you take the back or spine of your patch knife, or another tool, and tap along the edge of the flint to knock off some flakes to get your rifle or musket back into action. This is used in a shooting competition or in a battle reenactment.
Two, with the cock fully forward holding the flint, you use the edge of the frizzen where it contacts the back of the pan and press downwards against the top of the edge of the flint to cause it to flake. This can be tough on the cock and on the frizzen hinge.
Three, as you mentioned in the video, you either leave the flint in place, or some people remove it and put it on a hard surface, then they take a small, brass punch, and tap the butt of the punch with a small hammer while the tip of the punch is against the edge of the flint..., to make small flakes, and to sharpen the edge.
Four, you remove the flint, and place it on a thick piece of leather as a pad, you use the tip of a small piece of antler, and very carefully press down with that tip, against the edge of the flint. This is the ancient pressure-flake technique used to sharpen stone knives, spear points, and arrow points, and this seems to be the most controlled way to remove as little flint as possible to hone the edge, thus you get the most life from the flint.
LD