  We use state-of-the-art cosmological zoom simulations to explore the
distribution of neutral gas in and around galaxies that gives rise to high
column density \ion{H}{i} \mbox{Ly-$\alpha$} absorption (formally, sub-DLAs and
DLAs) in the spectra of background quasars. Previous cosmological hydrodynamic
simulations under-predict the mean projected separations $(b)$ of these
absorbers relative to the host, and invoke selection effects to bridge the gap
with observations. On the other hand, single lines of sight (LOS) in absorption
cannot uniquely constrain the galactic origin. Our simulations match all
observational data, with DLA and sub-DLA LOS existing over the entire probed
parameter space ($-4\lesssim $[M/H]$\lesssim 0.5$, $b<50$ kpc) at all redshifts
($z\sim 0.4 - 3.0$). We demonstrate how the existence of DLA LOS at $b\gtrsim
20-30$ kpc from a massive host galaxy require high numerical resolution, and
that these LOS are associated with dwarf satellites in the main halo, stripped
metal-rich gas and outflows. Separating the galaxy into interstellar
("\ion{H}{i} disc") and circumgalactic ("halo") components, we find that both
components significantly contribute to damped \ion{H}{i} absorption LOS. Above
the sub-DLA (DLA) limits, the disc and halo contribute with $\sim 60 (80)$ and
$\sim 40 (20)$ per cent, respectively. Our simulations confirm analytical
model-predictions of the DLA-distribution at $z\lesssim 1$. At high redshift
($z\sim 2-3$) sub-DLA and DLAs occupy similar spatial scales, but on average
separate by a factor of two by $z\sim 0.5$. On whether sub-DLA and DLA LOS
sample different stellar-mass galaxies, such a correlation can be driven by a
differential covering-fraction of sub-DLA to DLA LOS with stellar mass. This
preferentially selects sub-DLA LOS in more massive galaxies in the low-$z$
universe.
