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 andH(0)2.5 sigma lower than a hostz-only sample, and we explore several possible sources of bias that 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 higherH(0)than previous analysis which, for the Pantheon data set, could result in supernova and cosmic microwave background measurements agreeing on omega(m)h(2)despite tension inH(0). To obtain rigorous results, though, the Pantheon catalog should be improved by obtaining host spectra for supernovae that have faded, and future surveys should be designed to use host galaxy redshifts rather than lower-precision methods.