There are in fact many types of "noise" that are either not directly related to the signal or are correlated with the signal.
There is thermal noise, which is essentially white noise and cumulatively can be heard as the "hiss" one hears from many electronics. This is usually simple noise and not generally correlated with the signal in any way.
There is hum, which can be mechanical, electrical or both. This is seen as harmonics from the AC line frequency (60, 120, 180, 240 etc. in the US and 50, 100, 150, 200 etc. in Europe). This can be either simple noise and uncorrelated with the signal OR it can be correlated with the signal in some circuit designs. This can be clearly seen in an FFT scan where the harmonic distortion components are surrounded by a lot of "fuzz", which is modulation of the harmonic distortion components with the frequencies generated from the line voltage. So, from noise we get a new kind of distortion on the signal. Here is a clear indication of where power cleanliness is potentially very important to the sound we hear.
Look at this link to see what I mean by modulation of the harmonic distortion with power supply harmonics.
https://www.stereophile.com/content...n-mono-monoblock-power-amplifier-measurements
What you can see is that there is a regular series of line harmonics in the noise floor (Figure 3)
There is clear evidence of modulation in the 50hz FFT (Figure 8)
There is clear evidence of modulation in the 19+20Khz IMD FFT (Figures 9 and 10) below 5Khz...it is more evident at 1 watt where the high order harmonics have dropped to the noise floor. There is also a lot of fuzz around the 19 and 20 Khz peaks as well as the sidebands at 100 watts.
https://www.stereophile.com/content/audio-alchemy-dpa-1m-monoblock-power-amplifier-measurements
This one also shows intermodulation with the power supply harmonics.
There are even worse examples but I thought an example from a couple very modern designs would highlight that this is far from a solved problem in most amps.
Then there is RFI and EMI, these can also work there way into the signal chain and although I have not seen measurements showing their impact (like the line noise example) on audio, I have seen the effects first hand working with Mass Spectrometers and laser systems. Our laser system was a pulsed Nd:YAG system that had some ultra high speed switches. These switches and the high voltages controlled by them generated huge bursts of RFI that we were picking up on the coaxial (shielded of course) signal cables. Clearly the shielding they had was not adequate, which is why we were wrapping critical areas with aluminum foil and then running grounding wires from that foil. It didn't eliminate the problem but reduced it. So, I know these parasitic RFI and EMI signals can pollute your noise floor but even worse is when these things get modulated with the signal (like the hum example) and create more "noise", which is in fact a new kind of distortion.
There is also back EMF "noise", which is really a form of distortion but will only afflict amplifiers with substantial amounts of negative feedback. This was first pointed out by Matti Otala in a couple of papers in the 1980s. This is another form of signal correlated noise that then gets intermodulated with the music signal thus further reducing fidelity.
Finally there is negative feedback itself. Negative feedback reduces the levels of lower order harmonics and results in the generation of high order harmonics. This has been demonstrated by Baxandall, Crowhurst, Pass and others. The argument is that THD is reduced but as Crowhurst pointed out you get an endless array of high order harmonics, multiples of multiples of multiples. This creates a complex "noise" floor that is signal correlated and modulated. This means it is moving up and down. It is fundamentally different than the thermal noise or unmodulated hum noise, which are not correlated with the signal and you can actually hear correlated content BELOW a true random noise floor...what you cannot do is hear those very soft signals below a non-random noise floor because the correlation matters to our ear/brain pattern recognition.
IMO, what people talk about when there is no music playing and what they mean by a lowered noise floor when the music is playing are two quite different things. The noise with no music playing is the intrinsic noise in the system from thermal and hum maybe some RFI/EMI effects as well. However, when the music plays, you have potential intermodulation with all the noises listed above except perhaps the white thermal noise (maybe there as well). How well a circuit design rejects interferrence from power supply noise, RFI/EMI, back EMF will have a big impact on how low it's "noise" floor is and esepcially if there is a lot of negative feedback involved, which will make many of these intermodulation effects more rather than less likely.
I think when people hear a big benefit from a power cable that is good at rejecting RFI/EMI, it indicates that their electronics were being negatively impacted and a percpetion of a "blacker background" is one of the most commone benefits. I also think that this is one of the reasons good SET amps, if not pushed too hard, often do a better job of making a 3d image/soundstage because they are not being hampered by the negative feedback and back EMF issues (although they most certainly can have issues with the power supply intermodulation and RFI/EMI contamination).