Again?! 
With this particular Yo-Bwl, begin the extension the same, but take the end around
BOTH legs of the eye, and tuck it out through the central loop-nip (not the collar).
This meets your criteria, and throws a 3rd dia. of rope into the SPart’s nip, to soften
its curvature to boot. This is similar to what Wright &Magowan presented in 1928,
which is just slight change (as to which side end is tucked beside itself) from what
Heinz Prohaska presented ca. 1990 as a “dbl. bight Bwl”. [And what I show as the
“Doubly Tucked Half-hitch” would be what I’ve described above if the end dives
down between the other two parts in the loop.]
This URLink has been here before, but here’s another shot.
(trying http://i3.tinypic.com/wjwh1t.jpg
http://i3.tinypic.com/wjwh1t.jpg
One can do something similar with the Water Bwl, and if the water runs like
a Cow hitch vice a Clove hitch, the result is aptly named “Mirrored Bwl.s”
–which is my thinking re the “Janus” moniker: it’s a Bwl both coming & going.

And it seems that although some of these structures might loosen a little
they don’t loosen too much (or can be tied loosely w/o concern). The collar
around the eye leg can be (should be?) drawn tighter than the lookalike
main collar around the SPart; holding this leg in will prevent the entire body
of the knot from loosening upon the feed of SPart in through its collar, which
is to be expected, really (and which looseness enables the Bwl to be easily untied).
I like the way the Cowboy Bwl can be oriented, running end’s first pass after
making the SPart collar in so that the end-eye-leg is between it & SPart in
the central loop, making the eye-legs wrap, then tucking down between
the two parts already within the loop.
Esp. in stiff rope (such as some esp. aged caving/SAR low-elongation kernmantle),
the Yo-Bwl is difficult to form, as it asks for the end to make a 1-dia. turn around
the end-eye-leg (though one can swing wide and tuck out on the other side
of the SPart, to ameliorate this); and in forming the knot (indeed, as I just did
now, in aged BlueWater II rope), it’s not hard to pull the end wrap out of position,
resulting in the SPart making a 1dia. turn that is more towards the eye than that
of the end (which turn should be just within the eye)! This was a problem that
Heinz’s Nylon Highway article pointed out; and a token/example of this
misformed knot mistakenly appears on the Chapter 3 Knots cover page
in [u]On Rope (1st ed.) ! --and beside it is the form of the FIg.8 loop/bend
that Dave Merchant favors (well, the lower half is, but the upper half does
not have the same geometry). Hmmmm.
–dl*