If we wish to distinguish this bowline-like loop from other “similar” “proper” bowlines, we can say that it is a crossing knot - based eyeknot - its nipping structure is not the “common” single nipping loop of the “common” bowline, but a crossing knot.
Now, the interesting thing is that those knots, when capsized, can be transformed into “proper” bowlines, indeed - the “coming and going” bowlines, where the nipping structure is now simpler ( the crossing knot has degenerated into a single “proper” nipping loop), while, at the same time, the collar structure has become more complex ( a double collar, around the standing end / tail pair and the eye legs pair). Moreover, the Jug Sling / Mike s fancy bend - based bowline is also a TIB knot. See : http://igkt.net/sm/index.php?topic=4336.msg27161#msg27161
Do you prefer to tie it with an end, or in the bight ? To me, it seems easier to tie it with an end, as you show - while, for the pet loop shown in (1), it is easier to tie it in the bight.