Redshifts used in current cosmological supernova samples are measured using two primary techniques, one based on well-measured host galaxy spectral lines and the other based on supernova-dominated spectra. Here, we construct an updated Pantheon catalog with revised redshifts, redshift sources and estimated uncertainties for the entire sample to investigate whether these two techniques yield consistent results. The best-fit cosmological parameters using these two measurement techniques disagree, with a supernova-only sample producing $Omega_m$ 3.2$sigma$ higher and $H_0$ 2.5$sigma$ lower than a hostz-only sample, and we explore several possible sources of bias which could result from using the lower-precision supernova-dominated redshifts. In a pilot study, we show that using a host redshift-only subsample will generically produce lower $Omega_m$ and matter density $Omega_m h^2$ and slightly higher $H_0$ than previous analysis which, for the Pantheon dataset, could result in supernova and CMB measurements agreeing on $Omega_m h^2$ despite tension in $H_0$. To obtain rigorous results, though, the Pantheon catalog should be improved by obtaining host spectra for supernova that have faded and future surveys should be designed to use host galaxy redshifts rather than lower-precision methods.
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