what is the intended purpose of the white single turn?
In general, we should not suppose that all knots are "designed" according to a certain "plan", to satisfy some "intended purposes" !
Of course, after we have tied a knot, we can always ask ourselves if there are any redundant elements that can be subtracted / omitted, without making any important sacrifices.
This loop, in particular, was tied the way it is tied, just because of its resemblance with the original loop, the Lee Zep bowline. Without the second turn, it looks like another one of the many crossing knot loops we already have.
However, I believe that there are some advantages offered by the presence of the second turn, indeed. First of all, the collar follows a softer curve around the standing end, which, generally, is always a good thing. Similarly, the coil-like Eskimo-type continuation of the eye-leg-of-the-bight is following a three diameters path, which is also good thing. With stiff ropes, a less wide coil would tend to unwind itself.
Last, but not least, the very important subject of the stability or instability of the nipping structure. With "instability", I mean the tendency of the closed nipping loops to open up, and degenerate into open helices. I have seen that many double nipping loop bowlines are quite stable - and some of them will pass the "ultimum bowline stability test
" : their nipping loops will remain closed, even if the rim of their collar is cut off ! ( The three most stable double loops bowlines I know are the Water bowline, the Girth-hitched bowline, and the double crossed nipping loops bowline.)
This characteristic of the double nipping loops bowlines makes me appreciate the presence of the second nipping loop, even if it might look as a superficial/redundant element. I believe that, in a degree, the same happens in the Lee Zep Band B and C loops, too. Of course, the real gem of this thread is the original Lee Zep bowline - and I think that the "shining" it acquired in its X form ( where the tail goes under /crosses the other penetrating segment ) made it a most secure end-of-line loop ( that has also the great advantage to be bowline-like, and thus be able to be tied and untied in one step).
For some other crossing knot loops, see : http://igkt.net/sm/index.php?topic=3467.0
At the attached files see yet another simple crossing knot loop, posted elsewhere.