Isn't the angle of the point whatever the cutter wants it to be? What am I missing here?

The angle in the facets of the point has to be right to keep the light bouncing back out through the face and not right out the bottom, and it's determined by the physical properties of the material. For silicon carbide the index of refraction is higher, so the light will be bent more sharply when it enters, resulting in greater separation of white light into color components ("fire") and necessitating shallower angles in the point to maximize internal reflection