I have stated many times that MP does some things differently than traditional FPS's, but it's still a first-person shooter. Just because you can jump and solve puzzles in it doesn't mean it's not an FPS.
Well every FPS has some kind of 3rd person element in it, whether its in a vehicle or something like that.
When I played through MP I found it was at least 90% First-person, but I also did a ton of backtracking in that game, probably more so than most. Puzzles in that game just weren't obvious to me at all, I'd have to backtrack three or four times before what I was supposed to do became apparent.