Use "trailer" video instead of first video from TMDB
On TMDB there are different typs for videos you can mark them as Trailer, Teaser, Clip, Featurret. And this should be the priority order if there are several youtube videos.
Currently it seems to be random which vidoe will get displayed on letterboxd. See for example http://letterboxd.com/film/oldboy-2013/ currently you get a featurret if you hit play trailer even while there is a trailer on TMDB see: