So does this PLL have 2 modes of operation Locked Phase and Locked Frequency?
No, not quite. First, it does not make sense to analyze an out-of-lock situation, because that is chaotic.
When in lock, input and VCO frequency are the same, and the phase difference between them is constant.
The difference between PD I and PD II is, that for PD I, the phase difference is constant, but varies with frequency, as it needs to provide a PWM signal that can be filtered to a DC voltage for the VCO.
For PD II, the phase difference is always zero, and the output is three-stated most of the time. This is also one of the reasons why you need an active loop filter.
Please note, that for PD I to function correctly, the input must either be a sine wave or a 50% duty cycle square wave.