From the comments:
Why the power slightly goes down when the compressor is on? (I mean, look at the arc-shaped curve during the compressor cycle).
It must mean that it's getting easier to pump the fluid. That's a question for the mechanical guys, I suspect.
Why does the power factor spike to 1 exactly at the beginning?
It may represent that the motor is acting as a resistor at startup and it's inductance is playing a less significant role. It's a transient so it may not be all that accurate. I'd also check how much "headroom" the measurement system has: what is the peak current handling capability? how many samples per second can it handle?
The compr PF is only due to the charac of the compressor or there is more to it?
It will be determined by the characteristics of the motor. How do we know? Because the PF can be brought back to unity (1) by the addition of power-factor correction capacitors in parallel with the motor without affecting the operation of the motor itself (or its load).
Why during that compr phase the PF seems to be actively maintained to that value? Is it really like that?
It looks like the ratio of W:VAr is being maintained throughout the cycle. (Both curves have similar arcs). Therefore the PF is steady.
I've measured the idle power and it is 2W, 9VA, PF 0.180. Does it say something on the power supply? Why this low?
It doesn't say anything about the power supply - only the load. We can expect that the unloaded motor will run close to synchronous speed and back EMF will be at a maximum. Therefore the resistive losses will be low and the motor's inductance becomes more significant and the power-factor decreases as a result.