The published article annotated modes of failure with respect to the desired operational objective of a 5kton blast, 500 ton, various component failures, and their implications. It's written from a operational impact perspective on the war, Potsdam, and cold war implications.

Read the article, but he's talking about failures as in not exploding (or exploding with significantly lower yield).

Perhaps setting the atmosphere ablaze would have been an exceptional success? I remember someone telling me that this possibility was still being discussed before the Trinity test and, again, after thermonuclear bombs entered the arena. This seems to be a nice summary: http://large.stanford.edu/courses/2015/ph241/chung1/

I think they refer to the failure mode where: Nothing happened for a day or two. All the experts travel to the location to try to understand what went wrong, in situ... and THEN the explosion occurs.

This really wasn't a plausible failure mode. They had many (9?) detonators which had to be very precisely simultaneous, and had done a lot of tests of those alone (this was cheap & easy). As long as just one of them worked, you would burn all the explosives (just not in the right order) and destroy the device.

I'm sure they had some redundancy in the systems required to fire those detonators too, but how much?

Not sure how you would quantify this, but enough? Inventing detonators that worked fast enough (i.e. small enough error in firing time) was one of the challenges. They did a lot of tests of those explosive lenses. (32, not 9, of course.)

The electronics to fire them would have comparatively easy to test well, and I'm sure they over-designed it all. I doubt that duplicating it would have made anything better.