Two meanings involve the use of 'fail' in the sense of 'failure'. Of them, excepting the phrase 'without fail', as you've observed, the meanings are or were obsolete. Colloquial usage has reversed the obsolescence of 'fail' in the sense of 'failure'.
For the first, obsolete except in the phrase 'without fail', the meaning of 'fail' equivalent to one sense of 'failure' is this:
1. = failure n. 1. Obs. exc. in phrase without fail; now used only to strengthen an injunction or a promise; formerly also with statements of fact, = unquestionably, certainly. †Also, in same sense, (it is) no fail (but), sans fail: without any doubt, for certain.
The meaning of 'failure' referenced is this:
The fact of failing.
a. A failing to occur, be performed, or be produced; an omitting to perform something due or required; default.
The second sense of 'fail' equivalent to a sense of 'failure', and wholly obsolete until recently, is this:
2. The fact of becoming exhausted or running short, giving way under trial, breaking down in health, declining in strength or activity, etc.
As you've observed, this second formerly obsolete sense of 'fail' and the first formerly obsolete (except in the phrase 'without fail') sense of 'fail', both equivalent to different senses of 'failure', have been revived in informal usage.
The information at the site you cited was largely accurate, although it was incomplete and failed to provide any authority for the assertions.