  We study the field-induced magnetization processes of extended
Heisenberg-Kitaev models on the honeycomb lattice, taking into account
off-diagonal and longer-range exchange interactions, using a combination of
Monte-Carlo simulations, classical energy minimization, and spin-wave theory.
We consider a number of different parameter sets, previously proposed to
describe the magnetic behavior of $\alpha$-RuCl$_3$ and Na$_2$IrO$_3$ with
their antiferromagnetic zigzag ground states. By classifying these parameter
sets, we reveal the existence of three distinct mechanisms to stabilize zigzag
states, which differ in the sign of the nearest-neighbor Kitaev interaction,
the role of longer-range interactions, and the magnitude of the off-diagonal
$\Gamma_1$ interaction. While experimentally hardly distinguishable at zero
field, we find that the three different scenarios lead to significantly
different magnetization processes in applied magnetic fields. In particular, we
show that a sizable off-diagonal interaction $\Gamma_1 > 0$ naturally explains
the strongly anisotropic field responses observed in $\alpha$-RuCl$_3$ without
the need for a strong anisotropy in the effective $g$ tensor. Moreover, for a
generic field direction, it leads to a high-field state with a finite
transversal magnetization, which should be observable in $\alpha$-RuCl$_3$.
