The OP should look no further than his own question -- the word is gunfighter. The iconic unaffiliated hero of Western movies is Shane, and Shane was called a gunfighter.
Shane (Alan Ladd), a skilled, laconic gunfighter with a mysterious
past, rides into an isolated valley in the sparsely settled state
of Wyoming, some time after the Civil War. (emphasis added.)
Shane was one of the all-time great Western movies. The source linked above says:
Shane is a 1953 American Technicolor Western film from
Paramount, noted for its landscape cinematography, editing,
performances, and contributions to the genre. The picture was
produced and directed by George Stevens from a screenplay by A. B.
Guthrie, Jr., based on the 1949 novel of the same name by Jack
Schaefer. Its Oscar-winning cinematography was by Loyal Griggs.
Shane stars Alan Ladd, Jean Arthur (in the last feature, and only
color, film of her career) and Van Heflin, and features Brandon
deWilde, Jack Palance, Emile Meyer, Elisha Cook, Jr., and Ben
Shane was listed No. 45 in the 2007 edition of AFI's 100 Years...100
Movies list, and No. 3 on AFI's 10 Top 10 in the 'Western' category.
In his review of Shane, Roger Ebert called Shane a gunfighter:
Yes, on the surface, Shane is the gunfighter who wants to leave his
past behind him, who yearns for the sort of domesticity he finds on
Joe Starrett's place in the Grand Tetons. Yes, someone has to stand up
to the brutal Rufus Ryker (Emile Meyer), who wants to tear down the
fences and allow his cattle to roam free. Yes, Shane is the man--even
though he knows that if he succeeds he'll have to leave the valley.