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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.