I think Westerns are characterized by rough justice, revenge and vigilantism in the setting of a failed state(or a frontier that has a similar lack of good governance), where one is obliged to take the law into one's own hands due to the lack of althernatives. This equally describes the old west or the Japan of Kurosawa's movies, or other places. So the setting and the type of action in a western are intertwined, with the flawed hero being the lone gunman or sheriff for example, in an otherwise lawless place bringing justice.