  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.
