I replaced the PM100D power meter with another DET10N that I borrowed from Tanioka-kun, and I repeated the measurement.

This time the two signals at the oscilloscope look really similar. See the attached videos. (also the coherence on the spectrum analyzer is close to 1, sorry I didn't save the coherence data).

Then I closed the loop using the PD#1 in-loop and the PD#2  out-of-loop.
Then I exchanged them and closed the loop. See the two figures.

The control loop reduces a lot the noise in-loop but it doesn't really work for he out-of-loop PD (same situation when they are exchanged).

One possible reason could be the clipping noise, because I'm not sure how precisely the beam is focused inside the area of the PD. 

Another possible reason could be the OD2 filter (that I'm putting after the laser to limit the power and avoid the PDs saturate). If I remove it, 40mW would imping on each PD. I'm not sure how safe it will be for the PD, and In order to avoid saturation, I will have to drastically reduce the load resistance.

Another way to reduce the power would be to enlarge the beams up to much more than the PD size (which is 1mm).