I think it is more of rephrasing or broadening the definition of signal than the definition of noise, also being dependent on context.
Strict sense of signal
In a more strict sense, when we talk about SNR (signal to noise ratio), for example, then noise is not a signal, because "signal" is then closer to your definition, which is the relevant part of the broader signal.
Broader sense of signal
This "broader signal", from which noise is a part of, is what's been captured by a sensor, a measurement system, or is some general data transformed in some way to what we would call a signal (a numeric array of data that relates to one or more independent variables). However, if you're using a microphone to capture vibrations in air, even if it's "noisy" (e.g. it's random fluctuations), it's still what you're trying to capture from the environment in general (you're not getting temperature information, surface rugosity, light, etc.), so it would still be relevant for your purposes in a sense.
Supporting with a reference
I think the SNR part is already clear and more intuitive, but Oppenheim and Willsky's book may help to clarify what I mean in the third paragraph.
Translating from a portuguese edition of Signals & Systems:
"The signals, which are functions of one or more independent
variables, contain information about the behaviour or nature of some
phenomenon, while systems answer to particular signals, producing
other signals or some desired behaviour."
So if all you're measuring from a microphone is noise, the information it's carrying is that the sound behaviour (at that time and place) is noisy - and then this "noise" would still be a signal.
Checking the Wikipedia page on signal, in the first paragraph, we notice that the "it conveys information" role is not emphasized by all authors.
In the audio signal Wiki, it is simply "a representation of sound".
Alan V. Oppenheim, Alan S. Willsky. Sinais e sistemas, 2 ed. Pearson. 2010. ISBN 978-85-4301-380-0.