The song was correctly guessed by @wizrad@lemmy.ca: Johnny Guitar (1954) by Peggy Lee. This is the title track to the 1954 Western movie of the same name, starring Joan Crawford.

“Got a song for you, now. It’s about a guy who’s cold on the exterior, but deep down, you know he’s a good man. And his name… is Johnny Guitar.” -Mr. New Vegas (Fallout: New Vegas)