At least in Trevor Noah's case he grew up in South Africa where he was absolutely and objectively treated differently under law due to one of his parents being black. The fact that his other parent was white did not give him any advantages on that front. I imagine that colors his world view quite a lot.
The US used to be the same not that long ago. Didn't matter if both or only one of your parents where "black", you were equally banned from the 'whites only' parts of society. Even today in the US racists probably won't be less racist against you just because one of your parents are white.
the external aspect of that person that is defining the person and the color is taking precedence on everything
Basically yea, at least in the US. I think the best working definition you will find for 'black' is would the person have been kicked out of a "Whites only" establishment in the Southern US in the 50s. My gut feeling is that that is the definition most Americans instinctively use.