USB to RS485: Can I connect A to Common Ground?
It was a mistake. By connecting the dongle GND to the common GND on my board, instead of output B, not only the strong noise completely disapeared (meaning something there was conflicting tension between A and B) but an indicator led on the dongle stopped shining.
Because they sell these things without any serious instruction, I didn't know this LED means there is something wrong with the network. I thought it was normal.

I'm very suprised that my experiment worked because I thought the difference between between 2.3V and 4V would not be interpreted correctly by the schmitt trigger, but it was. The good surprise was to see a short drop to near 0V, for about 2us. This drop allows the schmitt trigger to go low and it returns to high only after its imput crosses the high level treshold.

However the schmitt trigger's high level treshold (typ 2.4V at 4.5V Vcc)  is very close to the low state voltage of output A (2.3V) and could even be under if lower than typical or Vcc is less than 5V.
So it's not very reliable IMO. 

This is the signal with the corrected circuit:
The yellow line is A output from the dongle.
The pink line is the signal after the schmitt trigger.