Yes, that's the original version and meaning. Recently it seems to have morphed into an either/or concept -- the mule gets the carrot for being good or gets hit with the stick if it is stubborn.
What makes you think the carrot will serve as motivation if it is never given once the task is achieved? Contrary to popular opinion, animals ain't dumb.
@ChipmunkErnie Yes, that references "dangling a carrot" - the saying that metaphors the promise of reward.
Where that saying comes from is "to use the carrot or the stick" - which metaphors reward and/or punishment. Orignially the rider held both motivators, not just one.
It is a saying/concept that is found in other languages and manners: "The pastry or the whip" "Five fingers or five bullets" "Pano palo" = bread or the stick Patronage and Fear Positive and negative reinforcement
