Two types of three-step chain-branching kinetics for gaseous hydrogen–oxygen detonation quenching are proposed. The reaction rates dependency on density and pressure, as well as the change in molecular weight are included and fitted to results of detailed chemistry not only along the Hugoniot curve but also behind transverse waves. The trend of induction and reaction times and reduced activation energy of zero-dimensional constant pressure combustion process as well as the induction, reaction lengths, and peak thermicity of the ZND model are focused on in the calibration procedure. One model (3SM-I) is designed to model the ideal detonation whereas the other (3SM-N) focuses on the $D_n$ - $κ$ curve during calibration, i.e. the detonation velocity-mean front curvature curves, aiming to capture the non-idealities. The predictive ability using the proposed models is evaluated in a detonation transmission configuration, a layer of mixture being confined by N2, in determining the critical height, reactive layer height at which detonation transmission is not possible. Both approaches get closer to the critical height from detailed chemistry. With the present set of model parameters, 3SM-I is better than 3SM-N for prediction of critical height. Moreover, borrowing analytical ignition criteria and wave hierarchy approach, the comparison of various chemical models that are able to recover the critical height suggests that the slope of the $D_n$ - $κ$ curve, linked with the chemical response to losses, is the first key parameter for the critical height determination. In addition, the dynamics of the extinction near criticality such as appearance of transverse detonation and quenching distance from 3SM-I are in very good agreement with that of detailed chemistry due to better estimation of the chemical features along the Hugoniot curve and behind transverse waves. The present study provides a guideline to get simplified chemical models from detailed chemistry for predictive simulations.