Okay, explaining the problem to someone (as often) helped, I stumbled over the solution just after posting :ROFLMAO:. For anyone else running into this, you need to enable "Full Kinematic Contact" to get "kinematic/kinematic + kinematic/static collisions" for the Cube... Enabling that and then...
I'm struggling with getting this exact same thing working, but in 2D with a non-trigger collision.
So taking the provided TriggerExample.unity as a startingpoint and modifying it:
case A (collision with non-trigger)
- Changing the cube-object to a RigidBody-2D, Kinematic and with a box...