"More formally, in statistics, a random process is a repeating process whose outcomes follow no describable deterministic pattern..."

But if it's a process requiring inputs, those inputs must be identical otherwise this doesn't apply. I'm saying that it's next to impossible to provide identical inputs into biological systems. There's far too much going on.

"But as you can see from the above the purely statistical definitions are not the only "right" ones."

That's right, and that's yet another reason for avoiding using the term to characterise the occurrence of mutations. One may become confused... ;-)