Relation between Normal Phase Shift of a wave and Phase Modulation
When adding a constant time shift, the equation is then $\cos(2\pi f (t + t_1))$ (note the parentheses around the time values).
$(xA_m)$ should really be $xm(t)$ in your phase modulation equation - it is a time varying quantity.
$xm(t)$ has units of radians.
It is 'phase modulation' because what is changing directly with the message signal is the phase of the transmitted signal. With amplitude modulation, the amplitude envelope of the transmitted signal carries the message; with frequency modulation, the change in frequency of the transmitted signal (as compared to the carrier) holds the message; and in phase modulation it is the phase of the carrier signal that holds the message. FM and PM tend to look similar because they both affect the argument of the sine wave, but they do so in two different ways.
It is not necessary to have several cycles of the carrier at any given phase shift in order to make that shift significant. Since we know the frequency of the carrier signal, that can be removed from the received signal to recover the message signal. Don't consider it from the perspective of a single time instant (as you point out, this is an analogue scheme, not digital), but rather consider operating on the signal as a signal.
"Now, how fast does the system notice the phase changes of the message signal?"


Continuously. That's how analogue systems work.