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(abridged) When stripped from their hydrogen-rich envelopes, stars with initial masses between $\sim$7 and 11 M$_\odot$ develop massive degenerate cores and collapse. Depending on the final structure and composition, the outcome can range from a thermonuclear explosion, to the formation of a neutron star in an electron-capture supernova (ECSN). It has been recently demonstrated that stars in this mass range may initiate explosive oxygen burning when their central densities are still below $ρ_{\rm c} \lesssim 10^{9.6}$ g cm$^{-3}$. This makes them interesting candidates for type Ia supernovae -- which we call (C)ONe SNe Ia -- and might have broader implications for the formation of neutron stars via ECSNe. Here, we model the evolution of 252 helium stars with initial masses in the $0.8-3.5$ M$_\odot$ range, and metallicities between $Z=10^{-4}$ and $0.02$. We use these models to constrain the central densities, compositions and envelope masses at the time of explosive oxygen ignition. We further investigate the sensitivity of these properties to mass loss rate assumptions using additional models with varying wind efficiencies. We find that helium stars with masses between $\sim$1.8 and 2.7 M$_\odot$ evolve onto $1.35-1.37$ M$_\odot$ (C)ONe cores that initiate explosive burning at central densities between $\rm \log_{10}(ρ_c)\sim 9.3$ and 9.6. We constrain the amount of residual carbon retained after core carbon burning, and conclude that it plays a critical role in determining the final outcome: Chandrasekhar-mass degenerate cores that retain more than $\sim 0.005$ M$_\odot$ of carbon result in (C)ONe SNe Ia, while those with lower carbon mass become ECSNe. We find that (C)ONe SNe Ia are more likely to occur at high metallicities, whereas at low metallicities ECSNe dominate.