  We model the isotropic depinning transition of a domain-wall using a two
dimensional Ginzburg-Landau scalar field instead of a directed elastic string
in a random media. An exact algorithm accurately targets both the critical
depinning field and the critical configuration for each sample. For random bond
disorder of weak strength $\Delta$, the critical field scales as $\Delta^{4/3}$
in agreement with the predictions for the quenched Edwards-Wilkinson elastic
model. However, critical configurations display overhangs beyond a
characteristic length $l_{\tt 0} \sim \Delta^{-\alpha}$, with $\alpha\approx
2.2$, indicating a finite-size crossover. At the large scales, overhangs
recover the orientational symmetry which is broken by directed elastic
interfaces. We obtain quenched Edwards-Wilkinson exponents below $l_{\tt 0}$
and invasion percolation depinning exponents above $l_{\tt 0}$. A full picture
of domain wall isotropic depinning in two dimensions is hence proposed.
