We show how spectra of Type Ia supernovae (SNe Ia) at maximum light can be used to improve cosmological distance estimates. In a companion article, we used manifold learning to build a three-dimensional parameterization of the intrinsic diversity of SNe Ia at maximum light that we call the Twins Embedding. In this article, we discuss how the Twins Embedding can be used to improve the standardization of SNe Ia. With a single spectrophotometrically-calibrated spectrum near maximum light, we can standardize our sample of SNe Ia with an RMS of $0.101 pm 0.007$ mag, which corresponds to $0.084 pm 0.009$ mag if peculiar velocity contributions are removed and $0.073 pm 0.008$ mag if a larger reference sample were obtained. Our techniques can standardize the full range of SNe Ia, including those typically labeled as peculiar and often rejected from other analyses. We find that traditional light curve width + color standardization such as SALT2 is not sufficient. The Twins Embedding identifies a subset of SNe Ia including but not limited to 91T-like SNe Ia whose SALT2 distance estimates are biased by $0.229 pm 0.045$ mag. Standardization using the Twins Embedding also significantly decreases host-galaxy correlations. We recover a host mass step of $0.040 pm 0.020$ mag compared to $0.092 pm 0.024$ mag for SALT2 standardization on the same sample of SNe Ia. These biases in traditional standardization methods could significantly impact future cosmology analyses if not properly taken into account.
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