How can be the comparison between stationary and white noise processes be discussed?

ANSWER

two great ways to think about stationarity, though infinitely more exist.

The first is about the notion of "the same distribution". All that means is that the process doesn't change (in a probabilistic way) over time: if we look at the process now, vs looking at it a million years in the future, it will have the same properties (mean, autocovariance, etc) which we can infer; it doesn't matter when we start.

The second way is through information. Everything you know about time T and time T+K is encoded entirely in their gap K. If you know how far apart two points are in time, you know their relationship.

For example, a white noise is stationary but may not be strict stationary, but a Gaussian white noise is strict stationary. ... Loosely speaking, if a series does not seem to have a constant mean or variance, then very likely, it is not stationary.