Everything looks good, your S51 you got looks better then the one I bought from that guy!

Mine was in an awkward position in that metal cage and some of the plates on the outside of the ballast have started to peel a bit. Still works great though.
I took mine out of the metal plates but surprisingly it didn't get much lighter, the metal plates are pretty light.
I can't quite see the mogul wiring in your pics but that part is common sense, white->white black->black