I bumped this up because I was finishing off a bimini twist and I remembered this from the Albright discussion on how to lock the tag end of an Albright:
He forms a small loop with the tag end and then wraps the tag around the main line inside of the loop he formed. Wrapping back towards the main Albright knot. Pulling the tag end to tighten the keeper knot back towards the Albright knot. The keeper knot will tighten up against the Albright knot. Trim the ends and you are good to go. This is how I have always tied it and don't ever remember it failing.
I realized that I start off the finish of the bimini exactly like that but add a different twist. A friend of mine who fishes a lot of Marlin off Equdor or Costa Rica or one of those places showed me how to do it. A deck hand down there showed him how to tie it, and would not let him use the ones he had done.
You do exactly as you described - form a "D" loop ( the straight part of the "D" being the standing line) and run the tag end inside the "D"around the standing line 4 or 5 turns or so back towards the Albright.
But rather than cinching it up, instead, you grab the new loop formed by the tag toward the end away from the Albright, and wrap it around the other direction, starting at the Albright end - kind of like doing a whip finish on a fly head without a whip finish tool. What you are doing is unwinding what you just did, but starting from the Albright end this time.
What you are left with is that the tag end is now INSIDE all the wraps PARALLEL with the standing line - with the end of the tag still going the same way, towards the Bimini (Albright).
To cinch it up, you have to have the standing line taunt and then keep teasing the wraps away from the tag end you are tugging on to tighten the knot, and then towards the tugging end as the loops get closer.
I started doing it this way for a couple of reasons. The most important is that it's almost impossible to burn the standing line. But also because all the wraps are around both the standing line, and the tag end , helping to cushion the standing line.
It is basically like a backwards nail knot and is the best way to finish off a bimini I've ever seen.
I rarely ever tie an Albright, but I'll do it that way if I ever tie another one. For Biminis , it's the cat's meow, and I expect it would be no different with an Albright.
Unfortunately, I've never seen this one published, named or videoed - but someone out there may have a name for it.