Do directors have a voice in choosing movie posters, or is it always a producer or external choice? Even for different movie posters in different countries, do directors ever decide to use a specific image for their movies?
I'd say that directors do not have much input into designing the poster, since they are usually made by third-party design studios, with more then one of these usually competing to make the best and be chosen by the marketing staff of the movie studio.
I'm a big fan of movie promos that used to be shown on TV. They were five to ten minutes or so, and gave the stars and director a chance to tell us how this was the best piece of art they ever been in . . . or something like that. Networks would broadcast them when a movie ran short, so if a ninety minute movie with John Wayne ran 1:50 with commercials, a promo of a new John Wayne movie would fill out the time slot, then two minutes of commercials, followed by the next show or late local news. Other promotional things might run, too. Not too often the director might talk about the poster and who did it. Never saw them claim any credit for promotional stuff, including poster.
I have seen interviews where directors claim they had no say or were forced to accept stuff for the sake of the movie. In "Heaven's Gate" the director Michael Cimino directed a cynical, self-indulgent take on a incident between cattlemen and sheep herders where one person died. The movie poster had the two leads, a flag, and the image of a man with a rifle claiming, "The only thing greater than their passion for America, was their passion for each other." There is no evidence anyone (characters) in the movie gave two hoots about America, including the director. If the Medved brothers are to be believed, the poster flowed from the studio trying to sell / salvage this movie--it didn't.
For most big budget flicks, the director makes the product, the producers and the studio package it. In smaller (budget) films, the lines are rather less formal, so the director may have other roles, including the poster. A powerful (money-making) director can have much more say over incidentals. Ultimately, no one person is ever in control, or you wouldn't have the possibly apocryphal translation of "Psycho" into "The Man Who Dressed Like His Mother."