What does "Dark Background " mean TT terminology

Justubes, there are in fact two different issues here, but my understanding is that "dark," what you call "first dark" refers to something's tonal attributes, as in: The current Levinson amps have a dark sound.

The other issue, noise floor, or background noise, what you call "second dark" I've always heard referred to as "black" or "black background", never "dark." As in: The sound emerged from a totally black background. I've never heard the term "dark" used in this context.

In the first case, a range or degree is implied (dark, darker, very dark ;--)

The second case implies an absolute -- absolute silence. The sound never emerges from a "dark gray" background ;--)
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You can also just invent your own terminology. It will be as useful as the terms many of the reviewers employ. Is your system too yang? Does it lean toward the whiter side of orange. Is there a self-effacing restraint in the upper registers as if the contrast was just slightly too sudden on light jazz at lower SPL? Have you found that Madonna displays a bit too much cleavage when entering contralto range?
See it's really quite easy. The real task lies in getting hired so that you can be paid to jibberize.

I think Johnnantais nailed it pretty good. I'm also not suprised that earlier audiophile definitions don't address this. It seems to me that the issue of noisefloors in systems is a more recent area of attention: for a long time, the issues were around tonality, range, dynamics, imaging and other attributes of the sound the system was making, as opposed to the background 'noise' that wasn't supposed to be heard. Not so much hum, or rumble, or even the level of ambient 'noise' the system is producing while on, when no program is playing.
Instead, as I think Johnnantais suggests, it is the difference between the music on the one hand, and utter silence on the other- the more of the latter, the more the music is going to emerge from an environment that is uncolored and contrasts starkly with the sound of the instruments. If the system has a 'sound' in place of this silence, there is not as much contrast between the music and the silence- so the small details are obscured; in addition,
the 'sound' of the system will also be present when notes are playing and color the music in the same way. So, to me, a dark background means dead quiet in the silences, and also speaks to the lack of a coloration being imposed on the music itself.
It may be that with quieter sources- CD perhaps, for the lack of surface noise, and improvements in electronics, as well as AC, we are hearing more artifacts of the equipment itself- not just what the equipment is designed to sound like, ie, its obvious colorations that are as much revealed in the silences between notes as the sounds of the notes themselves. The attention paid to this also seems to coincide with all the consideration now given to isolation stands, AC conditioning and the like.

The subtleties of the leading edge of notes, note decay, reverberant space, hall depth, etc., that contribute to a realistic rendering of music, seem to be revealed in contrast to dead silence (blackness). But it's debatable whether the absence of this desireable attribute is caused by a higher noise floor, or by false signatures imposed by lesser equipment at the threshold of audibility. For example, when the music isn't playing, a good SS system is generally quieter than even a great tube system due to tube noise. But to me at least, the tube system can be just as effective at communicating the subtleties of music on the verge of silence. In any case, the presence of modest tube noise in the background does not make it any less "black" sounding.