... this knot PART is IDENTICAL to an ABK knot part. There is no difference in dressing, certainly not before its loaded.
I do not believe that we can say this. The "half hitch" and the "nipping loop" may be considered as having identical "parts", but they are not the same. If "difference in dressing" means "difference in shape", without any consideration of what is going to be loaded, and how, I think this "difference" is not what makes knots be the same or be different ! Structures are not determined uniquely by their shape. A dome and a fishing net may have the same shape, but they are altogether different structures.
Besides that, the shape of the Butterfly loop and the shape of Mobius loop, are NOT the same ! Mobius should had presented his knot without this funny angle of the eyelegs - if he had done that, and the eyelegs were parallel or at, say, 30 degrees angle, the "difference in shape" would had been more obvious - but it would not be THIS difference which would be the main reason those two knots are different : the main reason is that they are loaded differently, and the various segments of the knot inside the nub "work" differently.
Xarax I don't know if you actually aren't seeing this or if you have some purpose in refusing to acknowledge a fact. There is no good reason raised at this point for there to be any disagreement about the facts of this knot part or these two configurations. Nobody disagrees that this knot part can contort and behave differently under different combinations of tension and angle applied to its 4 ends. A 2x4 can support much more force loaded axially than beam loaded, and acts sometimes as a stud or as a joist . A bolt is still just a bolt though if it shear loaded, or stretched, whether pre-loaded or not. Arguing in a serious way about labels is a fools game. I'd rather argue about facts. The loading facts are not particularly in dispute(not fully understood either, but not in dispute) as far as I can tell.
But NOW you've gone and seemingly intentionally failed to clarify, refused to establish, or if I give you the benefit of the doubt, are maybe simply wrong about a simple fact, either because you actually don't see it or because you have some purpose which the fact does not suit. While this might seem wise to you, dodging facts to me does not seem likely a useful way to find truth. I think it's much more helpful usually to establish facts than to argue about words.
The (false) fact I'm referring to the possible implication that this knot part, aside from loading or what exact angle we lay the ends at , is actually different, and that's just wrong. If the loop is cut and the 4 ends are all cut the same length you cannot distinguish one knot from the other. They are INDISTINGUISHABLE. This is a fact and not open to interpretation of the minds eye or to any number of words of extreme wisdom.
I have one of each sitting on each of my two legs. I tied them both as regular ABL's, but capsized
one of them to the other form to produce it in the "right" way. I have spread and opened each so that they clearly retain the essence of their dressing but are now easily vissible. One has the loop toward me the other has the loop away. They are now turn for turn, crossing for crossing, over for under exactly the same knot part. If I cut all the ends short, there would be no difference at all and they will then simply be exactly the same knots. I could take a picture, but what it would it prove? That I can tie the same knot twice? You have to do it yourself.
Without loading, this loop is just as happy to have its eye legs at the angle mobius shows as any other angle. Yes the "butt cheeks" of the loop legs tend to slide apart, but that's true with normal loop loading or ring loaded, so is irrelevant. There are actually at least a few dressings for this knot within either of the "two" capsized "variations". I don't think that changes their names, although I think it does change their properties, but if someone wants to give those different names too, I cannot argue factually against that. It's impossible.