Thanks for the picture, Burgo.
I think after our PM exchange we have both a clear picture what is going on.
My picture should have looked like this:
It was confusing that I indicated a swap of two +-centres relative to its neighbours. It is much clearer to show the real swaps on the white face, like in my new picture.
The picture of the inner 3x3x3 shows the hidden inner 3x3x3 edge swap.
Burgo and I agree, that the hint to the Real 5x5x5 was interesting but not really necessary. (At least it was confusing for me and may be for others, because I had no clear picture what was displayed by the circle pieces.)
I'll add some text later.
1. My sequence does the job
2. It is really not so easy to understand why
EDIT: I made a picture of the internal 3x3x3 Supercube. It is identical to Burgo's (as usual I use the Gelatinbrain style of front view / back view), just with Supercube stickers.
I still use lower case letters for the faces.
The turns Ll2 Ff2 U2 l2 U2 Ff2 Ll2 U2 effect the inner cube partially only, i.e. the lower case part l2 f2 l2 f2 l2 (translated to normal 3x3x3 notation L2 F2 L2 F2 L2) produce this result on a solved 3x3x3 Supercube.
Personally, I find that the paradigm of a 3x3x3 Supercube inside a 5x5x5 Supercube explains nicely why my sequence works. I view the pieces in the white and yellow layer as glued twins of pieces in the outer and inner layers. So, what we recognize with this mental picture is a 2-2 swap of 3x3x3 edges - one hidden - and a 2-2 swap of 5x5x5 +-centres.
I do not see the need to care too much about the hidden 3x3x3 edges swap: One is visible and mathematics tell us that there needs to be another one.
I was confused by the yellow pieces on the white face on the Real 5x5x5. (I did not understand what they are displaying.) This was the reason why I assumed that there was a misunderstanding. If anybody is interested, Burgo can explain nicely which pieces are displayed in the circles of the Real 5x5x5.
I had confused Burgo by my typo plus the misleading arrows in my X-Cube pictures
Burgo's (Ff2 Rr2)x3 does the job too: It produces 3 swaps of baby face edges on 3 different baby faces.
BTW, if the Real 5x5x5 becomes available ever in a physical form, I'll buy it in a heartbeat. And I bet, Burgo as well