  We present SAMI-HI, a survey of the atomic hydrogen content of 296 galaxies
with integral field spectroscopy available from the SAMI Galaxy Survey. The
sample spans nearly 4 dex in stellar mass ($M_\star = 10^{7.4}-10^{11.1}~ \rm
M_\odot$), redshift $z<0.06$, and includes new Arecibo observations of 153
galaxies, for which we release catalogues and HI spectra. We use these data to
compare the rotational velocities obtained from optical and radio observations
and to show how systematic differences affect the slope and scatter of the
stellar-mass and baryonic Tully-Fisher relations. Specifically, we show that
H$\alpha$ rotational velocities measured in the inner parts of galaxies (1.3
effective radii in this work) systematically underestimate HI global
measurements, with HI/H$\alpha$ velocity ratios that increase at low stellar
masses, where rotation curves are typically still rising and H$\alpha$
measurements do not reach their plateau. As a result, the H$\alpha$ stellar
mass Tully-Fisher relation is steeper (when $M_\star$ is the independent
variable) and has larger scatter than its HI counterpart. Interestingly, we
confirm the presence of a small fraction of low-mass outliers of the H$\alpha$
relation that are not present when HI velocity widths are used and are not
explained by "aperture effects". These appear to be highly disturbed systems
for which H$\alpha$ widths do not provide a reliable estimate of the rotational
velocity. Our analysis reaffirms the importance of taking into account
differences in velocity definitions as well as tracers used when interpreting
offsets from the Tully-Fisher relation, at both low and high redshifts and when
comparing with simulations.
