Your first sentence is correct. Your second sentence is not. It *is* the case here. 0dB is not "no sound". 0dB (SPL) is "a standard sound". 0dBm is the equivalent of a 0.632 Volt peak-to-peak sine wave signal into a 50 Ohm load (1 mW). 0dBW is 1 Watt. etc,
dB is usually for power, so in those terms:
relative power (in dB) = 10*Log(actual power/reference power).
For zero power:
relative power (in dB) = 10*Log(0) = - infinity
Doing it backwards:
actual power = reference power * 10^(relative power (in dB) / 10)
For a relative power of 0dB:
actual power = reference power * 10^0 = reference power.
For a relative power of -infinity dB:
actual power = reference power * 10^(-infinity) = 0.
For sound, dB can also describe pressure. Depending on the reference (pressure or power), 0dB corresponds either to an (rms?) pressure of 0.00002 Pascals or 10^-12 Watts or 10^-12 W/m^2 etc.
0.00002 Pascals corresponds supposedly to the softest sound the average human can hear, but it is not "nothing". By this definition about half of humans could hear it. But maybe "0dB" is marketingspeak for "noone can hear it" (technically not true of a 0dB sound), as opposed to "no sound". If you look at a VU meter or a mixing desk or Cubase or audio software with sliders (measuring electrical signals in dBu here), 0dB is loud. The bottom of the sliders are "- infinity."
0dB implies that there is no change from the reference sound. -infinite dB would imply that you are decreasing the reference sound by an infinite log base 10 amount, which is not the case here.
So in this case it seems, to me at least, that 0dB is indeed correct as that component doest increase or decrease the intensity of the sound compared to the reference.
Am I right?
First a small correction, dBm is 1mW into a 600ohm load not a 50ohm load. This would give a voltage of 0.775V. This is also known as dBv or dBu when dealing with professional audio electronics.
Second, dB can be used to describe any ratio of two numbers. The user just has to specify the reference used and what is being described, ie Sound Pressure level, sound power level, voltage, current, etc.
0 dB SPL (sound pressure level) is what is called the threshold of hearing. It is the softest absolute sound that can be heard. If you have two sound source both producing a sound at the threshold of hearing, 0dB, the total sound pressure level would actually be 6dB SPL. 6dB SPL is twice as loud as 0dB SPL. And 0dB SPL is half as loud as 6dB SPL. As a note, from psycho-acoustics, the perceived doubling or halving of SPL is 10dB.
Here is some reading on sound pressure.
Wiki on Sound Pressure
In this situation, "0dB Cooling" means no added sound to the environment. Using this terminology was most likely a marketing decision not an engineering decision, because saying something has "-infinite Cooling", even though it is the proper terminology, could seem to imply, to the average consumer, that the cooling solution is also a noise canceling device, which is not the case. The companies probably don't want the trouble of confusing their average customers with too much engineering speak.
As for the dB scales on mixing consoles and audio software, the references are different than the reference for dB SPL. -Infinity on a console or software means that none of the input gets transfered to the output, while 0dB means 100% of the input signal gets transfered to the output.
-Infinity dB = 0V while
0dB = 1V for a system referenced to 1V
0dB = 0.775V for a system reference of 0.775V
etc...