Relation between Normal Phase Shift of a wave and Phase Modulation I am little confused with the Phase Modulation and the phase of a sine wave. I get the phase modulated wave from the google images as below:

Phase modulation has the form:
$$PM(t)=A_c \cos\big(2\pi f_c t+(xA_m)\big)$$
Considering the message signal as
$$ m(t)=A_m\cos\big(2\pi f t\big)\text,$$
and $x$ as the phase modulation index.
Basically, I see $\cos(\theta)$ denotes a wave with some frequency. If we consider in this form - $\cos(2\pi f_1 t)$, it is a cosine wave with frequency $f_1$. And if we add a constant time factor with the unit of seconds, we get something like: $\cos(2\pi f_2 t+t_1)$. I am able to see this as a right phase shifted cosine wave with time $t_1$.
But during phase modulation, we actually change the phase of the wave as per this form:
$$PM(t)=A_c\cos\big(2\pi f_c t+(x A_m)\big)$$

Now what I am confused is that the term $(x A_m)$ has the unit of radians or seconds?
Based on the figure, I am able to assume $x$ has the unit of radians/Volt making the unit of $(x A_m)$ as radians, so that the final phase modulated has an abstract form $\cos(\theta_1+\theta_2)$ which could be some other frequency of the original carrier wave. I am just guessing this but I am not sure. If you could please let me know if this assumption is valid, it would be helpful.

If it is valid, then why do we call it as a phase modulation?
For a phase modulation, I see that the modulated waveform should be something like - $PM(t)=A_c\cos(2\pi f_c t+(x A_m))$ where $x$ should be in the unit of (seconds/volt). Now the phase of the carrier wave changes but I am not able to imagine the waveform.

This does not exist but isn't this the pure phase modulation?


Now, the next hardest part which I am concerned is that, I see that phase modulation is an analog modulation scheme. Let's assume that at a time instant $t_1$, the Phase modulation system calculates the amplitude of the message signal and changes the frequency of the carrier wave proportionally. At time instant $t_2$, this frequency changes to the next value depending up on the message signal.

Now, how fast does the system notice the phase changes of the message signal?
Is that given by the carrier frequency or something else? (Since it is looking the amplitude of the message signal at a specific time instant, can we see it like -the message signal is somehow sampled, its amplitude is quantized to some constant (having the unit of radians) and then added to the angle of the carrier wave?

The reason why I am curious about this is that we can see the phase modulation as the frequency of the carrier wave is changed but that frequency should remain hold at least for one cycle to make it significant.
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.