  Absorbers in the spectrum of background objects probe the circumgalactic
medium (CGM) surrounding galaxies, but its physical properties remain
unconstrained. We use the cosmological hydrodynamical simulation TNG50 to
statistically trace the origins of HI Ly-$\alpha$ absorbers around galaxies at
$z = 0.5$ with stellar masses ranging from 10$^8$ to 10$^{11}$ M$_\odot$. We
emulate observational CGM studies by considering all gas within a line of sight
velocity range of $\pm 500$ km s$^{-1}$ from the central, to quantitatively
assess the impact of other galaxy haloes and overdense gas in the IGM that
intersect sightlines. The impact of satellites to the total absorber fraction
is most significant at impact parameters $0.5 R_{\rm vir} < b < R_{\rm vir}$
and satellites with masses below typical detection limits ($M_* < 10^8$
M$_\odot$) account for 10 (40) per cent of absorbers that intersect any
satellite bound to $10^{10}$ and $10^{11}$ $(10^9)$ M$_\odot$ centrals. After
confirming outflows are more dominant along the minor axis, we additionally
show that at least 20 per cent of absorbers exhibit no significant radial
movement, indicating that absorbers can also trace quasi-static gas. The
metallicity of absorbers also depends on the azimuthal angle, but this signal
is largely driven by enriched inflowing and quasi-static gas. Our work shows
that determining the stellar mass of galaxies at $z_{\rm abs}$ is essential to
constrain the physical origin of the gas traced in absorption, which in turn is
key to characterising the kinematics and distribution of gas and metals in the
CGM.
