Glad you fellas found this useful, as that is what this forum is all about!!
As for 4 jaw chucks, I have never seen a 4 jaw self centering chuck, but have heard mention of them from time to time. It might be that someone was asking if they exist, but I don't remember for sure. I am not saying that they do not exist, only that I've never seen one. Guess I was just never in the right place at the right time!
Four jaw independent chucks don't need to be set up to run true like 3 or 6 jaw chucks do, or any kind of self centering chuck, as they do not need to run to such close tolerances. When using a 4 jaw independent chuck, if you need a work piece centered, you always have to use a dial indicator to dial in the part. If you are turning something offset, such as an eccentric for a steam engine, the chuck does not need to run true then either, you still have to dial in the offset using a dial indicator. Thus, there is no need for a true running back plate on a 4 jaw chuck. The only thing on a four jaw that needs to run true is the face, it needs to be square to the spindle.
The practice of machining a back plate to run true on a particular lathe is necessary with faceplates also, especially on lathes with threaded spindles. Chucks and faceplates on machines with tapered spindles and cam lock spindles should at the very least be checked also, and will usually require a cleanup pass in order to run true when moved to a different lathe.