Hm, I am not convinced. What is the difference between a clockwise twist and an anticlockwise twist using your 2-cycle?

And what happens, if you do three times the same twist? This should be equivalent to no twist - but three 2-cycles cannot cancel.

Although I did not claim it to be a proof, below is an image I made which illustrates what went through my mind when I posted, at least.

As far as the = (rotate) sign, let's look at the counterclockwise rule (in the image above), for example.

Let 1 = the white sticker, 2 = the red sticker, and 3 = the green sticker on the solved cube to the left. We have the list {1,2,3} when the corner is solved.

For the counterclockwise rule, we say we have {1,{2,3}}.

**Interpretation 1**
When we want to swap either 2 or 3 with 1, we have {{2,3},1} = {2,3,1}, which is a 3-cycle.

**Interpretation 2**
When we want to swap either 2 or 3 with 1, we have {{2,3},1} = {1,{3,2}} = {1,3,2}, which is a 2-cycle. (When you swap two objects, there isn't a rule that says that you cannot change the orientation of one or both of the objects: as long as you exchange their locations, the swap is valid.)

That is, as you can see after the = (rotate) sign in the counterclockwise example, we can either view us swapping the green and red stickers (flip the 1x2 bar of the green and red sticker horizontally) and then swapping it with the white "rectangle"/square (flip the two bars vertically) (2-cycle),

or

we can view it at rotating both rectangles 180 degrees (3-cycle).

I'm not disputing your mention of the 2-cycle being repeated 3 times is not the identity permutation, but we cannot rule out "interpretation 2" just because it "behaves" like a 3-cycle since when either the clockwise or counterclockwise rule is applied three times, it returns the stickers to their original state. (If this was easy to see, I'm sure someone would have posted it earlier.)

Lastly, when I mentioned that we cannot view a corner twist as a 3-cycle (which is incorrect from one interpretation), I had in mind that since a 3-cycle is a composition of two overlapping 2-cycles, then if we can do a 3-cycle, we should also be able to do just a 2-cycle. However, I just realized that when qqwref mentioned that a quarter face turn does a 2 4-cycle of edge stickers, we cannot do one 4-cycle, as this destroys edge stickering. (In short, disregard that statement.)