Nice job taking the videos! You're seeing the computer perform a Power On Self Test, then hand off control to the boot device. Since the boot device (the PRISM card) is removed it doesn't find one and stops.
At the bottom of the POST screen during boot it says "Press DEL to enter setup". That's how you get into your BIOS. I would do that using your friend's computer and take a picture of every setting. Then go into the BIOS on your computer and compare the settings just to make sure they are the same. Markmon mentioned this back in post 17. It's worth trying.
When your computer does not boot, we still don't know why. When it does nothing it could be bad memory, processor, or other motherboard problem. At least if it starts booting and freezes you can rule out the video.
I would blow all slots with dry air to remove any dust. I would try 1 stick of known good memory in one slot, then try the same stick in the other slot.
How much memory did you buy? Your test is only showing 8192k (and your friend's), which is 8mb. Back in post 12 it was mentioned they were using a 128mb stick. Your BIOS will show how much is installed.
P.S. In this photo you posted there are two sticks of memory. One of them isn't installed all the way. You have to push down until those arms on each end click into place. Or is that an empty slot? I can't quite tell.