Which takes us back to the beginning.
You don't know what randomness means. You think it's a "thing". You said it was "behavior". But it's not. Not at all. It has a precise mathematical definition, and you should learn what it is so you stop abusing language.
A random variable is an OBSERVATION. Nothing more, nothing less. It is not a "thing". It doesn't speak to the "properties" or "behavior" of any system. It only speaks to YOU, while you're looking at it.
Randomness MEANS uncertainty. To YOU, the observer. That's all it means. It means YOU can not predict an outcome.
The generator of the outcome that you're observing, can be stationary, non stationary, linear, nonlinear, fractal, chaotic or anything else. If YOU can't predict it, it's random.
Math tells us HOW MUCH you can't predict it. There are MEASURES of your uncertainty. There are many such measures, many types and sizes of yardsticks.
Math also tells us how much you can INFER about the generator by observing it, but it does NOT tell us anything about the underlying mechanisms. That's Kolmogorov's Second Law, you can't see behind the veil, you can INFER but you can not define.
Precision in language is important. Look here, Wikipedia or any other encyclopedia will tell you straight up:
randomness is the apparent or actual lack of definite
pattern or
predictability in information.
[1][2] A random sequence of events,
symbols or steps often has no
order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable
You see? That's it, that's all. It doesn't say anything about the generator, it only says YOU can't predict it.
It doesn't say there's no memory, nor does it define generators or distributions. All it says is YOU can't predict it.
We can INFER distributions by making enough observations, and in some cases we discover the distributions obey equations. What we OBSERVE though, are moments. To the extent we can map moments to equations, we can INFER mechanisms.
If you try to step outside of these bounds you're violating the math. Chaos says nothing about randomness one way or the other. Randomness is an OBSERVATION, not a system behavior. System behaviors generate outcomes, which you then observe, and if you can't predict them, they're random. End of story.
To say that a distribution "predicts" randomness is completely false. It's an abuse of both math and language. Distributions define EXPECTATIONS, which is not at all equivalent to random variables. Shannon and von Neumann both defined the level of "surprise" which is the difference between your expectation and the measured outcome. Renyi took it a step further, he studied the yardstick and came up with some relationships between the measurements and the levels of surprise.
Do NOT abuse math and language by equating the observation with the system. They're not the same thing.