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The Radak in Sefer HaShorashim (see p. 245) on the root שבט seems to use the two words interchangeably but he does write -
"ונקרא הראש והמושל שבט כי דרך המושל לחיות שבט בידו ועוד כי בידו שבט מוסר העם שבט מושלים"
"And he is called the head and the leader of the tribe because it is the way of a ruler to live with a stick ('shevet') in his hand, ...
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