Well that's interesting. Thanks for responding to my post. If you agree with me, why did you name uniform distribution as requirement number 1? Is that just part of the "official" academic definition or something?

I am not able to motivate further, it's just that usually when we talk about random numbers we mean uniformly distributed random numbers. It is not a requirement in a strict sense. It's just the easier way to tell about randomness. We all have an intuitive understanding of rolling a dice or flipping a coin.

As a fun example, I used to run load tests for several systems using random numbers distributed under a power law.