SN 2002cx-like and SN Ia-CSM objects show similar early spectra and both belong to a young stellar population, suggesting that they could share the same progenitor origin. Adopting the framework of the common-envelope-wind model developed in Meng & Podsiadlowski, we propose that both subclasses of SNe Ia are caused by the explosion of hybrid carbon-oxygen-neon white dwarfs (CONe WDs) in single-degenerate systems, where SNe Ia-CSM explode in systems with a massive common envelope (CE) of similar to 1Me, while SN 2002cx-like events correspond to those events where most of the CE has been lost in a wind. Using binary-population-synthesis calculations, we estimate a number ratio of SNe Ia-CSM to SN 2002cx-like objects between 1/3 and 2/3, consistent with observational constraints, and an overall contribution from hybrid CONe WDs to the total SN Ia population that also matches the observed number from these peculiar objects. Our model predicts a statistical sequence of circumstellar material density from SN Ia-CSM to SN 2002cx-like events and normal SNe Ia, consistent with existing radio constraints. We also find a new subclass of hybrid SNe that share the properties of Type II and Type Ia SNe, consistent with some observed SNe, which do not have a surviving companion. In some cases, these could even produce SNe Ia from apparently single WDs.