Update: I've asked the Musician's Friend staff in their online chat thing if the "regular" LP Jr. has a nitro finish, and how many pieces of wood their bodies have, and they said the finish is nitro indeed, and the bodies are 1-piece bodies as far as they can tell (but I suspect that the man was being A. a salesman, and B. just estimating by looking at the guitar - so it might have just been one with a nicely matched woodgrain).
I've also asked the Thomann (because I live in Europe so Musician's Friend is a source of information only) support about the finish for their "regular" LP Jr, and they've replied (and I quote): "No NITRO!". Pretty confusing stuff, right?
That being said, I congratulate the lucky "regular" LP Jr owners who are happy with their purchase, but buying one sight unseen looks more and more like a real gamble to me at this point.