There is
wiki for the curious.
The wiki stuff of course is totally right but does focus too much on steady sinusoidal signals which show infinite time characteristic.
In practice we have to deal with more "realistic" signals which begin at a distinct time stamp and do stop sometime.
As the signal is not "infinite" in time (neither for past nor any future consideration) we have to consider additional artifacts which come along with the start and stop of any signal.
I think the most common term just to call those artifcts by name is transient noise. Actually its an additional wide broad frequency "content" that adds to the start of a signal (and also this happens again when the signal does stop, or does fade out).
In System theory science we have got the step response that provides us with good information about the way how a signal turns in and also turns out whenever a signal passes a transfer channel.
At least in Germany these things are called "transient response".
Fortunatelly there is LTspice which helps to "demonstrate" what is going on under the hood
in the time domain.
-6dB/octave, phase shift equals -45 degree
Frequency 100Hz, time cycle equals 10msec
The first cycle looks noticeable distorted.
At 10msec of the time base (remeber 10msec fits to 1 cycle at 100Hz) the phase shift equals already -45 degree.
A trained eye may notce some (additional) lowish frequency content for the first couples of msec that overlays the (original) sine shape
Sample for 1 time cycle, just to demonstrate what happens with transient response if the signal does "hard stop"
btw the time base 0..1msec might look like some "delay" but its nothing but distortion to the (original" signal shape caused by the transient response.
In "rough" easy understanding words: actually a negative magnitude of phase, for example -90 degree of phase "tries"
immediatelly to shift a sinoidal signal to become a (-1)cosine signal
Same -6dB/octave filter alignment but for "reverse" frequency response
Phase shift to the signal equals +45 degree
same situation as above the first cycle looks distorted but, in a different way than above.
At 10msec (after 1 cycle period is gone) it can be easily seen that the signal is phase shifted by +45 degree on the time base line.
The signal shape at 0..1msec looks like as if the signal "intends" to go back in time.
That's not surprising cause positive magnitude of phase shift "tends" to shift a sinoidal waveform to become cosine "shaped"
Transient response for the time domain
Thank's LTspice.
Hope its at least NOW understandable that phase on its own does never cause delay.
Of course phase deviation due to dfferent travel path in air is totally different animal.