Answer: Something which gets old school horsemen rolling their eyes.
The term "carrot stick" was invented by the Parellis, who promote a particular style of horsemanship (some people love it, some people think it's absolutely awful and teaches people to be afraid of their horse).
What is a carrot stick actually?
It's just a whip. And if it has a lash on the end, that's a "savvy string."
Really. The only difference is that, yes the Parelli "carrot stick" is more expensive.
(In other words, the horse world is as vulnerable to fads and gurus as anything else).
(You might detect some bias against the Parellis. They promote not wearing a helmet so you pay more attention. I have little patience with that).