Conceptual Questions on Colored Noise Process
Once you talk about the Spectrum of noise / process you implicitly says it is stationary in the wide sense.
What does it mean to have a signal with uncorrelated samples? Do you understand it means you can't using linear predictor, to have any information from all past samples ion the current one? Usually this is not how signals behave. This is exactly why correlated noise is hard. It behaves like a signal hence it is harder to separate them apart. One way to deal with the model of Signal + Colored Noise is to apply whitening transform on the data, apply White Noise methods and then apply the inverse transform.
Any process where the probability for a specific value depends on the index value.
Here are some specific examples (From Investopedia - An Introduction to Stationary and Non-Stationary Processes):



Noise is used when dealing with signals. We call noise to the non deterministic added (Usually, we have multiplicative noise as well) term of the signal. Since we have some index (Time / Spatial, etc...) for the signal we have it for the noise as well which means it is a process.