I am having a problem with my rear derailleur. I installed as per instructions, adjusting the H and L limits to what I believe is proper. The problem I am having is that when I am in 7th gear, shifting down doesn't actually shift. The cable tightens, but it just pulls the derailleur back instead of sideways along to the next gear. I have tried both tightening and loosening the cable in various amounts. I also noticed that I can push the derailleur from side to side (as if it were changing gears) while the cable remains at the same tension.
As this is the first derailleur that I have installed myself, I really have no idea whats wrong. Did I install it wrong? Is it broken? I'd appreciate any help.
EDIT Added a newer video that hopefully shows the problem better.