Personally, I don't find the phrase 'fully illuminated “dark side”' to be a convincing alternative to the physically more accurate term 'far side'. Of course NASA has only just emerged from the Earth's dark side as I write this (UK here, mid-morning), so I'm not expecting an immediate response from them.
And yes, I do know that 'side' is itself not entirely accurate because of libration [0] but that's a different hill to die on.
The far side is the darker side, though, at lunar night. Poetic proof: "The Earth shine might illuminate the light side of the Moon a little during the long night" (from Jules Verne, All Around the Moon https://www.gutenberg.org/cache/epub/16457/pg16457-images.ht...)
The far side is darker during lunar night (lit by starshine only; Full Moon on Earth) than the near side during lunar night (New Moon on Earth), because it receives both star- and max. Earthshine.
I'm not sure about Crescent Moon though: that only narrows the brightness gap slightly, right? Or I’ll have to ask if there’s an astronomer on board our flight.