Understanding Working of High-Pass RC Filter
Yes, you are correct. For better understanding I recommend this link:

https://www.electronics-tutorials.ws/filter/filter_3.html

Although I would like to add that Voltage across the capacitor is always a phasor perpendicular to the Voltage across the resistor. So you can imagine a right angled triangle with the base being the V across R and the perpendicular being V across 1/(frequency times C). Initially, at $\omega$ = 0 or small (in the stop band), the perpendicular is infinitely long hence the angle that the hypotenuse makes with base is close to 90 degrees. As omega takes a finite value that is not too large(in the transition band) the length of perpendicular decreases, and the angles reduces to an acute angle. Finally, as omega becomes large (in the pass band), the angle gets close to zero. The angle is what we call the phase and its variation is the phase response.