What is the mathematical relationship between phase modulated (PM) signal and the complex phase/magnitude of the frequency component at the carrier frequency used?
The phase of the carrier wave is your data-carrying signal. It also is the phase in complex equivalent baseband.
So, that's the mathematical relationship: equality.
Magnitude doesn't matter to PM....
As explained in Laurent's answer, including the last point, which equals the first point, just gives twice as much weight to that point compared to all the others. This doesn't explain a phase shift in your approximation. If you do things right you actually get an almost perfect fit, even with the last point included:
t = 0:0.15:1.5;
y = [2.200 1.595 1.031 ...
First (wrong) answer (for integrity) The $y$-value of the last point
is the same as the first one. As you apparently know the frequency,
this point comes in excess of the "fundamental period". It sounds like
this additional point comes like an implicit double-weight to the
first point of the period.
Second take: I have tried to fit the data, with ...