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I don't understand what that means. It sounds like something I would call "noise" (="variation which I do not care about") which is a quite a different concept from "random".
There is also "true" randomness, e.g. radioactive decay, which doesn't seem to be related to whatever I might care about. And if you put yourself into the paws of Schrodinger's cat, you might care a great deal about that trigger which breaks the poison vial, but does that make it not random?
As you yourself point out that's entirely circular and, besides, I have no idea what "idealized randomness" is.
You're basically talking about randomness as that which lies beyond the limits of (current) knowledge. Didn't you just come back to randomness meaning "unpredictable"?
Wouldn't idealized randomness mean utter lack of causality?