The so-called "cowboy bowline" should avoid the latter,
and is a Lapp bend (reverse sheet bend on ring-loading
( I have used the term "common bowline" - or "bowline* - for both loops, the "left-" and the "right-" hand bowline - in contradiction to the "Eskimo bowline" . )
In the case of the common bowline, I do not believe that the "reverse Sheet bend",- indeed
any bend where the one link is a loose nipping loop - can hold, when the standing end is not loaded at all. On the contrary, in the case of the "Eskimo bowline" and a "proper" ring-loading condition, the normal Sheet bend will hold. The difference you mention, between the two forms of common bowline, does not exist under those extreme circumstances you have described, because in that case
both loops will not hold ! Would you trust a "reverse Sheet bend" that is the only thing left there, after a gradual untensionong of the standing end and loosening of the nipping loop? It will not be a Lapp bend you have tied and tighten, it will be a Lapp bend that
should have been tied and tighten
by itself...or else, a Lapp bend that you
should have prayed to be self-tied, self-tighten and holding - if that would be of any help... Noope, if by "ring loading" we mean a completely loose standing end, we should not trust any form of the common bowline - and we should use a secure Eskimo bowline instead. ( In general, I use an Eskimo bowline when I anticipate that the angle between the eye legs would become greater than 120 degrees).( See the attached pictures for a secure Eskimo bowline).
Try the Rusty s loop : I believe it is a very nice, compact and quite secure loop, that is not known and used as much as it should. I do not like very much it as
a bend ( two loaded limbs )- because I think it is a little unstable, - but as
a loop ( three loaded limbs ), it is much better. You may also try your Sidewinder bend, in its untucked variation (1). Any similar elongated bend, where both tails leave the knot s nub from its middle, can form a loop suitable for "proper" ring loading ( where one of those tails takes the place of the standing end of the loop).
1)
http://igkt.net/sm/index.php?topic=2694.msg17078#msg17078