Ok, here is the latest. I ordered the new electric water valve that you posted and it came in today.

(They looked almost identical, except the new valve had a longer mounting plate, and some kind plastic shroud around the top)
I installed it, crossed my fingers, it still leaks! : ) I read all your replies again. I could never get to that valve in the basement to check to see if it was a drilll type or self piercing. I had called the previous owner who built the house months ago, and was told it was a drilll type.
I removed some of the drop ceilng panels to try to find this thing to see for myself. Also, I noticed a KINK in the line, right before the shutoff valve. I cut out the kink, and made the line go straight into the valve:
This seemed to fix the leak, which led me to believe this kink caused the issue. So, as a test I then put the old electric valve back in, and it leaked again. So, I thought maybe it was a combination of the kink and the old electric valve. SO, lastly, I put the new electric valve back in, and now it's still leaking again! I checked all connections, swaped them both out a few times, and it's still leaking. Argh.
Here you can see what happens after filling a big glass of water:
video I made of leak, .mpg
I really don't know how to tell if the valve in the basement is a self pieceing or not. Maybe these pics will help you. Also, I'm on a Well here, and the well tank has 25 gpm written on it, which I thought was pretty good, as most people in this neighborhood get 8-15.
Suggestions?
Also, here is a pic of the back of the fridge.
