I would say that it is the idiom "none the wiser" that you are thinking of. This means
to still be confused about something even after it has been explained
to you
I would then surmise that the authors of the sources you have given have simply substituted other adjectives into this saying.
This Ngram shows that "none the wiser" has been used significantly more than "none the richer" or "none the healthier", but that "none the richer" actually predates "none the wiser" in print. I can't speak for whether "none the + comparative" was once a standard form, but certainly it only really caught on with "wiser".
Nonetheless, with reference to the common idiom, we can assume that these other uses mean "still not [something], even after [something]."
Your third quote, in context, supports this:
it is clear that the medical establishment was unable to help their
condition and after expensive tests and manifold prescriptions, the
individual returns home poorer and none the healthier
That is, they are still not healthy, despite having sought treatment at a medical establishment.
With an adjective that takes "more" in the comparative, rather than an -er ending, we would of course then have "none the more", eg, "none the more beautiful".
To address your queries: it is not the same meaning as "far from", and is nonstandard outside of the idiomatic "none the wiser"; as such, it is best used in informal contexts.
Best Answer
I think this really boils down to a question of when you use no and when none.
There will be no less than sixty people at the gathering.
Few people in Britain drink Bovril these days, none less than me
Editing later:
The usage to which you further refer is highly idiomatic, and it is perhaps worth considering for a moment what it is saying. It doesn't mean that nobody of any significance less than the President was there. It means that one of the people there was no one less than the President.
Personally I would prefer to say no less than... but as none is short for no oneI don't think it really matters.