I'd be interested to hear what you think the difference is and how that applies to crypto.
Okay, you said:
Random is NOT the same thing as "evenly distributed" or "looks disordered".
Just because something is not "evenly distributed" and does not "look disordered", it does not mean that it isn't unpredictable. The quantity of data points matters: the frequency distribution smooths over time for randomness (guaranteed), not necessarily for predictability (it can, but that's not guaranteed).
To clarify, an even distribution can be predictable, an uneven distribution can be predictable, "looking ordered" can be predictable and "looking unordered" can be predictable. An even distribution can be random, an uneven distribution cannot be be random, "looking ordered" cannot be random and "looking unordered" can be random.
Frequency distribution is neither negatively nor positively correlated with predictability, while randomness correlates very well with frequency distribution.
For key/salt/modulus generation we want unpredictable numbers, not just random numbers. This is why my original post said "not merely":
As far as cryptography goes, we want unpredictable numbers, not merely random ones.
(PS, as you can see from some of my posts over the last ten years or so, I work in encryption, and delivered a postgrad encryption course (amongst others) at a university when I was still in academia). I'm not totally clueless yet :-)