You apparently don't think there is a difference between a broken tooth and a broken jaw, and you're asking me if *I* believe what I write?
Obviously, no the Rangers aren't looking for an LTIR exemption for Stepan given there is no LTIR in the playoffs. But you asked if medical reports go to the league. The answer, in general, is yes. In this specific incident, given the league was looking at supplemental discipline, one would have to be pretty naive (or ridiculously biased) to think that the league failed to request details of the injury to Stepan.
Also, the "upper" or "lower" body injury is for public consumption only. You can't seriously think that the NHL would have been satisfied with "he's got an upper body injury", can you?
Actually, scratch that. I do believe you could seriously think that.