  We explore the interplay between different definitions of distortion for
mappings $f\colon X\to \mathbb{R}^2$, where $X$ is any metric surface, meaning
that $X$ is homeomorphic to a domain in $\mathbb{R}^2$ and has locally finite
2-dimensional Hausdorff measure. We establish that finite distortion in terms
of the familiar analytic definition always implies finite distortion in terms
of maximal and minimal stretchings along paths. The converse holds for maps
with locally integrable distortion. In particular, we prove the equivalence of
various notions of quasiconformality, implying a novel uniformization result
for metric surfaces.
