Let me start by saying that I think it's really cool that you are showing us what you're doing via video. Even though some of the earlier ones were a little shaky, it's still a great way to get your point across. I really like it.
Your arm appears to me to be a little worn, but I can't tell for sure - I think I am more SUSPECTING a worn arm because of how loose is was. If the hole is not round 360 degrees at any one depth, it's worn. A quick check with the inside jaws of a calipers would confirm this real quick.
If the hole is not round, I'd either replace it, or, if you don't want to spring for a new arm, buy a pack of JB weld and coat the bore of the arm with the stuff before you put the ball joint on. The JB weld will fill the gaps and should hold just fine if done properly.
You didn't say how you got the old nut off, but if the new nut is a locknut, you'll most likely have to put the new nut on the same way you got the old one off, by putting some weight on top of the arm while you tighten the nut. Make SURE the nut bottoms out and gets tight! If it continues to spin, it's not right!
If you JB weld the arm, and the new nut is a locknut, I'd use a standard nut to attach the ball joint until the JB weld cures, then take the nut off and put on the locknut. If the new ball joint came with a castle nut, then you should have no problem bottoming it out.
After you finish your repair and use it for a while, check the tightness of the nut daily until it stays tight.