The assumption in your first post is wrong, the voltage is of course reduced for k < 1.
There a different methods to describe coupled inductors and it's important to clarify which model you are using. A straightforward model, that's often used in EM analysis of arbitrary conductor geometries is an impedance matrix:
Zij = Rij + ωLij
Vj = Ii*Rij
You feed a current into branch i and get a voltage at branch j.
For the simple case L11 = L22, you get L12 = L21 = k*L11.