carrot on a stick/the carrot or the stick
Authoritative dictionaries agree—the expression refers to offering to reward a stubborn mule or donkey with a carrot or threatening to beat it with a stick and not to a carrot being dangled from a stick. For me, the clincher is that no one actually cites the form of the “original expression.” In what imaginable context would it possibly be witty or memorable to say that someone or something had been motivated by a carrot on a stick? Why not an apple on a stick, or a bag of oats? Boring, right? Not something likely to pass into popular usage. This saying belongs to the same general family as “You can draw more flies with honey than with vinegar.” It is never used except when such contrast is implied.