Would your definition exclude The Proposition and Quigley Down Under, then, since they're set in the Australian outback?
In that case, your definition is too limiting to be useful. In reviews, those films were referred to as "Westerns." In advertising, the iconography of the Western was emphasized (with each film belonging to a separate cycle of Westerns). In the video store the movies would be found next to other Westerns. On IMDB, the genre listing is "Western."
If a Western has to take place in the American West, then The Magnificent Seven
doesn't really qualify, either. The Mexican villagers briefly cross the border to recruit some American gunslingers, but 90% of the proceedings take place in Mexico. Or do Westerns set mostly or entirely in Mexico get a pass because of the country's proximity to the West?
is on the right track with his discussion of syntactic and semantic elements of genre. Rick Altman's book Film/Genre
is a pretty smart book on the subject, and not too