Carrot and Stick

Which is it? The one on the left, or the one on the right?
I hear the phrase used many times a week, and almost universally the speaker means it in the sense of the picture on the right--beating the donkey with the stick and offering it a carrot in order to get it to move.
Isn't the left-hand picture more elegant? Doesn't it make more sense, really? Isn't the object to get the donkey to move? I agree that the people who refer to carrots as rewards and sticks as punishments are consistent in their logic, but I think that they have imported a foriegn meaning to the original sense of "carrot and stick."

Of course I have no proof that I know the original sense, but let's look at the internal evidence of the phrase itself:

1) We talk about "the carrot and the stick." Is this not an apparatus? A carrot to reward and stick to punish are separate things, the operation of one independent of the other.

2) Which brings up a question? Why do we need to beat the animal at all? Won't it be lured forward by the promise of the carrot? As long as the carrot is just out of reach, the donkey will continue walking, in anticipation of reaching it. The stick-as-fishing-pole, with a carrot dangling tantalizingly out of reach, provides the sort of continuous promise of reward the situation calls for.

3) Can one person both beat the rump with a stick and feed carrots up front? Perhaps by holding a stick with a carrot dangling from it, as in the first picture, while simultaneously beating the animal's behind with another stick, would work--but then we're both violating elegance and returning to the question of why we need to beat the creature in the first place.

4) Although it is a donkey, and stubborn--which is probably why people assume we have to beat it to get it to move--there is no more guarantee that the animal will move with beatings than it will without them. We have the carrot, after all.

5) Someone could argue that the permanent withholding of the carrot will eventually result in the donkey stopping out of frustration. Then, beatings would have to be administered or the carrot given to the donkey. Fair enough--give the donkey the carrot. Then put another one on the string and start the process over. This is no different from the feeding of the carrot-as-reward--there are still no beatings necessary, and I would be willing to bet that we save money on carrots.

6) Yes, of course the dangling carrot method is manipulative. You might even try to tell me that the reward-carrot/punishment-stick method is more honest or some such nonsense, but this will remain a difference of opinion between us. Sure, I prefer a little quiet manipulation to violence. I prefer not to beat my donkey.

7) And then there is the self-sufficient--and almost pastoral--beauty of the dangling carrot method. The fragmentation of effort, the modern specialization required to mount the reward-carrot/punishment-stick method really turns me off.

8) But enough of opinions. Which is the correct meaning of the phrase, "carrot and stick"?