That's incredible.! I have been using these off and on for 50 years and never new there was a difference. A look at:
https://www.mcmaster.com/thread-cutting-dies says it all.
The configuration of the die does not determine what use the die was built for doing. You must look at the geometry of the die threads where they will be cutting (or not cutting) the work. Threading dies have sharp edges and positive rake for removing metal as chips. That is how you make a thread where there wasn't one before. Dies designed for re-forming existing threads have negative rake, to push the metal back to where it belongs. The last thing we want to do is to remove metal when cleaning up a damaged thread, which weakens it. The method used to drive the die does not tell us the intended use, it tells us what setting it was designed to be used in. We do threading and cleaning threads on lathes, mounted in vises, in tight places like automotive engine compartments, and many others. The method of holding is only about what will work best to turn the cutting tool for the job at hand. Square, round, hex, or whatever, is about how the tool is going to be held to turn it.
Now, with all that having been said, it is quite common for round dies to be for cutting. Hex dies are more commonly used for repairing threads in automotive applications, where access to the work can be maddeningly difficult at times. So that is what we see the most. But common uses do not tell you what we have at hand. Get a magnifying glass and look at the geometry of the tool.
Taps have exactly the same confusion. Most, by far, are for cutting, and some few are for straightening existing threads. The only way to know is to look at the geometry of the tool.
Often, when you think a random die or tap you are using is dull and worn out, it is actually for re-threading. Magnifying glass!
It is also entirely possible to change one style to the other in a pinch, with a Dremel tool if necessary. It is fussy, especially when the tap or die is tiny, but you can get those tools to work the way you want them to for the job at hand.
As always, we need to be smarter than the tools, and inquisitive about how they work...