The white push button switch triggers the staircase lights. I close the circuit with a small reed relay. As I couldn't make any sense of the wiring, I added a "rogue" wire from the doorbell switch. It provides 13.8V AC with enough current capability.

The installation manual of this com system is quite terrible for people wanting to understand how it works. It only contains wiring examples for all the different setups you can choose from, but nothing substantial as to what the wires may do (which is power, which is signal...).