The Outside to inside kick is like the front snap kick, but starts out at an angle to the outside, crosses over in front of you, and then comes down. Inside to outside has the reverse direction.
I usually hit with the ball of my foot. For me the target is hold in front of me at about waiste height and I hit it at the top. For the higher belts it is held higher. The master describes it as a way to slap somebody on the face with your foot, but I can't kick that high yet.