Clarification concerning power spectral density
Considering you are using PSD and not BiSpectra implicitly impose you assumed your process is WSS (you only consider the delay between samples and not their absolute time regarding the time origin).

Considering WSS assumption your process' ensemble mean is constant and doesn't change with time (E[x(t)]=Constant). So the only difference between co-variance and auto-correlation is a DC bias which only affect your DC component of your PSD so for zero mean processes(E[x(t)]=0) there won't be any difference between these definitions.

Considering practical estimation of PSD always there is a large DC component in PSD which always leaks to other frequencies and mask them out. So in practice they usually remove the DC component (removes a DC bias from the process then calculate the PSD).