Ice nucleation in clouds is often observed at temperatures >235 K, pointing to heterogeneous freezing as a predominant mechanism. Many models deterministically predict the number concentration of ice particles as a function of temperature and/or supersaturation. Several laboratory experiments, at constant temperature and/or supersaturation, report heterogeneous freezing as a stochastic, time-dependent process that follows classical nucleation theory; this might appear to contradict deterministic models that predict singular freezing behavior. <br><br> We explore the extent to which the choice of nucleation scheme (deterministic/stochastic, single/multiple contact angles θ) affects the prediction of the fraction of frozen ice nuclei (IN) and cloud evolution for a predetermined maximum IN concentration. A box model with constant temperature and supersaturation is used to mimic published laboratory experiments of immersion freezing of monodisperse (800 nm) kaolinite particles (~243 K), and the fitness of different nucleation schemes. Sensitivity studies show that agreement of all five schemes is restricted to the narrow parameter range (time, temperature, IN diameter) in the original laboratory studies, and that model results diverge for a wider range of conditions. <br><br> The schemes are implemented in an adiabatic parcel model that includes feedbacks of the formation and growth of drops and ice particles on supersaturation during ascent. Model results for the monodisperse IN population (800 nm) show that these feedbacks limit ice nucleation events, often leading to smaller differences in number concentration of ice particles and ice water content (IWC) between stochastic and deterministic approaches than expected from the box model studies. However, because the different parameterizations of θ distributions and time-dependencies are highly sensitive to IN size, simulations using polydisperse IN result in great differences in predicted ice number concentrations and IWC between the different schemes. The differences in IWC are mostly due to the different temperatures of the onset of freezing in the nucleation schemes that affect the temporal evolution of the ice number concentration. The growth rates of ice particles are not affected by the choice of the nucleation scheme, which leads to very similar particle sizes. Finally, since the choice of nucleation scheme determines the temperature range over which ice nucleation occurs, at habit-prone temperatures (~253 K), there is the potential for variability in the initial inherent growth ratio of ice particles, which can cause amplification or reduction in differences in predicted IWC.