It isn't hard to define randomness. It's an epistemic condition on the knowability of Y given X.

Non-determinism is an incoherent definition of randomness; classical physical processes are entirely deterministic.

The point of a coinflip being random is that it is random with respect to the information both observers of the coinflip have. It isnt random with respect to /any/ piece of information.

There are almost no processes which are non-deterministic in this sense. Not enough to bother calling them random; and in physics we do not: the word is indeterminate. Randomness has nothing to do with quantum mechanics; it wasn't invented in the 1920s. It's an epistemic condition.

The RANDOM variable X, st. X ~ N(mean, std) provides a random number x -- x isnt random with repect to the outcome which produced x; nor is it random with respect to an index of a vector in which it is contained.

This does not match the intuitive notion of randomness though.

I would say for a given variable to be random, it must not be predictible, given any other variables that humans can know.

I don't think it makes sense to say that X is random "with respect to Y", that's just the definition of independence.

And a constant variable is independent from all other variables, but it's definitely not random.

Then only quantum indeterminate phenomenon is random, and nothing else.

I dont think many people have thought enough about the world to appeal to their intuitions.

Randomness isnt quantum indeterminacy.