Integral over power spectral density
Summation or integration of the spectral components yields the total power (for a physical process) 


yes, that's how "power density" is defined ...


  or variance (in a statistical process)


For any bound density, that's true.


  But would the total power correspond to the uncentralized second moment, so variance + mean²?


Yes, but if you have a process with a non-zero mean, that would imply that the spectral power density at $f=0$ diverges (a "Dirac delta $\delta(f)$" with "infinite height"), and that would imply the PSD isn't actually a function (it's still a distribution / test function, but seriously, this is just getting into math). 

If you will, just imagine your "mean" is something at very low frequencies ($f\in[-\epsilon;\epsilon]$), but not concentrated to exactly $f=0$.

Still works out the same, for anything technical (because nothing will ever experience whatever process they're observing for eternity, so who's to say whether something happens at $0\,\text{Hz}$ or $10^{-24}\,\text{Hz}$?).

Honestly, the wording of the article isn't the greatest.