How do I sharpen my circle cutter
The point leads. Otherwise, the tool will simply skate over the material. 

As you indicate, it should be to the inside of the circle to produce a clean disk, or to the outside of the circle to produce a clean hole. 

To sharpen, file or grind the entire end, keeping the face at same angle. 

I hold mine at an upward angle against my fine grinding wheel with the point corner upwards. 

Addendum: I know you didn't ask this, but while we're on the subject of circle cutters, I find that the imbalance of the spinning tool causes the drill to cut sideways with its flutes, enlarging the center hole and losing position. For this reason, I don't use a drill in the center of the cutter. I drill the pilot hole first and then use a smooth shaft of the same size.