On light twins, it is “dead foot, dead engine” (which rudder is not needed to maintain direction of flight). On airliners, it begins with maintaining direction, WITHOUT concern with which engine failed, until above a critical altitude, where a discipled two-pilot confirmation of which engine failed (engine instruments) process correctly identifies which engine has failed to prevent this very occurrence.
So, the confusion comes when “little airplane” pilots fly big airplanes, and forget big airplane training and disciple, and instead doing stupid things.