I just watched your videos. The "side-to-side" movement you commented on in your first video and observed on both sides is NORMAL. However, the looseness in the RH end that you showed in your second video is NOT.
I'd like to see another video like you did with the second video, but this time, angle the camera so we can see the bottom nut. I can almost guarantee you that this nut is moving back and forth as well. The nut SHOULD move as one with the arm, locked down to it, not rock back and forth. Compare this with the LH side, it is NOT moving I'm sure.
I'm guessing that your ball joint is just fine, however you lost the taper fit between the stud and the arm, and now the hole in your steering arm is egged out.