After watching both videos, I believe that whatever bent the rear derailer also bent the derailer hanger, which has your rear derailer out of alignment.
The rear derailer/derailleur is that complicated part you replaced that moves the chain between gears on the rear, and also keeps the chain from getting loose and falling off.
On most bikes, the rear derailer isn't bolted directly to the frame of the bicycle. There's a small piece of metal bolted to the frame of the bicycle, and the derailer bolts to that. That's a removable derailer hanger. Typically the hanger is a softer metal that will bend easier than the frame. It's done that way to help protect the frame from damage if something happens to the rear derailer. Some bikes that hanger is built right into the frame and can't be replaced.
You probably need to bend the derailer hanger back straight. Even if you replace it, new ones often need a small amount of alignment adjustment (bending). Because it's a small piece, it's difficult to bend it back straight just perfectly. In a shop, they'd use a hanger alignment tool which is basically a strong bar that bolts to where the derailer bolts into the hanger, has alignment gauges that are compared against the (trued) wheel, and can be used to bend the hanger back to the correct alignment. You use a large adjustable wrench, like this answer suggests, to bend it straighter, but it's likely you'll have trouble getting it quite as perfectly aligned as your other bike.
Here's an article on aligning the derailer hanger: http://www.parktool.com/blog/repair-help/rear-derailleur-hanger-alignment
While the misaligned derailer indicates a bent hanger, it's also likely that your derailer high and low adjustment screws aren't set right, and/or that the cable tension is off. Those things should be adjusted after the hanger is straightened (so that the derailer is aligned perfectly like on your other bike). The adjustment screws on the derailer are small screws you use to adjust how far it will shift in either direction. The cable tension slowly goes bad over time as the cable housing compresses ("cable stretch"), and can be easily adjusted with a small twist of a barrel adjuster on the cable.
Here's an article on adjusting almost everything else on the rear derailer: http://www.parktool.com/blog/repair-help/rear-derailler-adjustments-derailleur
