Instructor can't talk when H10-13.4 is plugged in
So after Juan's help and also talking to Becker I think I've found the answer: 

It's an unlucky impedance combination.

That means maybe I can get it to work by lowering the VOX-level (increase mic bias) far enough so my instructor can activate it but if I lower it too much, my own mic will constantly activate the intercom, resulting in constant noise.

Becker told me, this is likely due to an old wiring version being installed where both pilot and copilot's mic are connected in parallel to the same amplifier. So when my mic has a 'better efficiency' i.e. better conversion from mechanical swing to voltage, the VOX adapts the threshold (or as Becker said the bias voltage for the mic) for this and the amplitude of the co's mic is to low to activate it. 
A fix would at least require rewiring of the system to have dedicated amplifiers and bias supplies for both the pilot and co-pilot.

I think I confused some things in the physics here; if you point me to it, I will gladly correct it.

In general Becker's advise was to always use the same kind of headset for both pilot and all passengers in order to ensure best results (all volumes will be the same for all people then).