  We continue our investigations of the magnetorotational collapse of stellar
cores discussing simulations performed with a modified Newtonian gravitational
potential that mimics general relativistic effects. The approximate TOV
potential used in our simulations catches several features of fully
relativistic simulations quite well. It is able to correctly reproduce the
behavior of models which show a qualitative change both of the dynamics and the
gravitational wave signal when switching from Newtonian to fully relativistic
simulations. If this is not the case, the Newtonian and the approximate TOV
models differ quantitatively. The collapse proceeds to higher densities with
the approximate TOV potential allowing for a more efficient amplification of
the magnetic field by differential rotation. Sufficiently strong magnetic
fields brake down the core's rotation and trigger a contraction phase to higher
densities. Several models exhibit two different kinds of shock generation. Due
to magnetic braking, a first shock wave created during the initial centrifugal
bounce does not suffice to eject any mass, and the core continues to collapse
to supranuclear densities. Another stronger shock wave is generated during the
second bounce as the core exceeds nuclear matter density. The gravitational
wave signal of these models does not fit into the standard classification.
Instead it belongs to the signal type IV introduced by us in the first paper of
this series. This signal type is more frequent for the approximate relativistic
potential than for the Newtonian one. Strongly magnetized models emit a
substantial fraction of their GW power at very low frequencies. A flat spectrum
between 10 Hz and > 100 kHz denotes the generation of a jet-like outflow.
[Abstract abbreviated]
