Since the discovery of the unusual prototype SN 2002cx, the eponymous class of Type I (hydrogen-poor) supernovae with low ejecta speeds has grown to include approximately two dozen members identified from several heterogeneous surveys, in some cases ambiguously. Here we present the results of a systematic study of 1077 Type I supernovae discovered by the Palomar Transient Factory, leading to nine new members of this peculiar class. Moreover, we find there are two distinct subclasses based on their spectroscopic, photometric, and host galaxy properties: SN 2002cx-like'' supernovae tend to be in later-type or more irregular hosts, have more varied and generally dimmer luminosities, have longer rise times, and lack a Ti II trough when compared to SN 2002es-like’’ supernovae. None of our objects show helium, and we counter a previous claim of two such events. We also find that the occurrence rate of these transients relative to Type Ia supernovae is 5.6$_-3.8$$^+22$% (90% confidence), lower compared to earlier estimates. Combining our objects with the literature sample, we propose that these subclasses have two distinct physical origins.