I understand what you're saying but I don't think they can be put in the genre of the teen movie just because teens can watch them or are in them.
I saw Dead Poets Society as a teenager too (and also loved it - still do), but I still don't think it really fits in the category.
Also, if your theory for LOTR is tested further then that would makes Star Wars a teen movie - or any of the James Bond films!
To most people, teen movies are things like Ferris Bueller, The Breakfast Club, Weird Science, Sixteen Candles, Say Anything, Diner, The Outsiders, TAPS, Risky Business, YoungBlood, Class, Animal House, Bring It On, Can't Hardly Wait, Mean Girls, American Pie (although they're also in the 'gross out comedy' subset IMHO) etc etc