Most golfers know that chip shots stay low and pitch shots go high. What they don't know is what has to happen during the swing to produce such different shots.
I think of the chip as a left-arm swing, and the pitch as a right-arm swing. Try the drills above to isolate the feel. For chipping, where the swing is short, grip with your left hand only (top row). Take a narrow stance, play the ball off your right instep, and lean a little left. Then swing back and through feeling as if you move the whole club--handle to clubhead--in tandem with the swinging of your left arm.
For pitching, where you need a longer swing, grip with your right hand (bottom row). Take a slightly wider stance, play the ball in the center, and set your weight 50/50. To swing, let your right arm fold going back and straighten through impact. This produces more of a releasing action, with the club sliding under the ball and pulling your arm and body with it.
Notice in the impact shots above how the shaft is in line with my left arm on a chip and my right arm on a pitch. Feel that, and you've got it.