Single Side band Phase noise
This is where things get ugly:

Can you define what "one-sideband noise power in 1 Hz bandwidth" is?

In my world, it'd just be the PSD of that signal multiplied by 1 Hz. But then, there'd be no need to call it "one-sideband noise power"; it'd just be the noise power density in 1 Hz.

So, "customary" might very much be a "relative" statement. I'd expect that term to only show up in literature that tries to explain signal properties based on a real-valued, analog technological background (maybe trying to bring the complex signal's view to an audience of people that learned the real-valued signal theory first).
 Frankly, I find it an elegant move that much of the modern literature starts right off with complex signals, and just shows that a simple multiplication with a complex sinusoid can move it to baseband, and then introduces the "normal" (as in: non-quadrature) mixer based on that understanding.

This is probably where that "upper and lower sidebands overlap" comes from:

If you take a noisy bandwidth $B$ of passband around a carrier at $f_0$ (you observe $]f_0-\frac B2; f_0 + \frac BS]$), and mix it down using a quadrature mixer (giving you I and Q signals, which we commonly interpret as real and imaginary part of a complex-valued signal), you get signal in $]-\frac B2; \frac B2]$. 
The PSD of noise in that range then is the same as the PSD of noise in the passband. That happens because, mathematically, the quadrature mixer just multiplies with $e^{-j2\pi f_0}$, leading to a "clean" shift of the positive frequencies down to baseband (by $-f_0$).
If you, however, just multiply with a $\cos(2\pi f_0)$, you'll mathematically shift the positive to baseband, but also you shift the negative frequencies, which are a mirror of the positive frequencies, up (by $+f_0$).
Now you have the upper sideband being overlaid with the mirror of the lower sideband.
So, looking at this differently, this is an effect of using real-valued signals to represent something that would become complex baseband when mixed with a quadrature mixer.

Note that the statement is obsolete (or plain wrong), because frankly, unless you really want this for some reason, measurement devices either


mix onto an intermediate frequency (IF), so that none of the signal "crosses" 0 Hz when being downconverted, or
employ quadrature mixers, where that overlap doesn't happen.


Update I'll have to revise my above statement about the obsolence of the statement, because your book,


  Phase Noise and Frequency Stability in Oscillators by Enrico Rubiola


is a book on mixer theory, where you of course analyze the workings of the elementary mixers (which can be combined to form e.g. quadrature mixers), so that the author inherently works with the more "physical" real-valued signals, and then he's obviously right:

if you're characterizing the mixing element itself, you can't tell noise above and below the LO apart.