Good fundamentals are important to putting, but a basic understanding of how the arms and body should move is even more important. Many players get stiff, freezing the joints in their arms, which causes the body to move excessively during the stroke. The feel you want is actually more like pushing a broom: Your right elbow pistons back along your side as you draw the broom back, then your right arm extends downward as you push the broom away. The same is true to a smaller degree as you swing your putter. This allows your body to stay quiet while your arms swing freely.
On a high pitch shot, the goal is to keep the bounce on the bottom of the club exposed to the ground throughout the swing. Players get into trouble when they turn the right arm over in an aggressive release. The feel you want is the same as the one you'd get skipping a rock across water. Your right arm throws from the side, fully releasing your wrist and elbow while keeping the palm of your hand facing the sky, or underneath the grip, as you turn through the shot.
It's hard for the average player to resist the urge to try to lift the ball on a chip, or to hold the face open to create loft. Imagine a Ping-Pong paddle in your left hand. The face of the paddle represents the face of the club. Instead of scooping or holding the face open, picture the face of the paddle turning to the ball, like a topspin backhand -- the secret to crisp contact on chip shots.