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Ice nucleation plays a pivotal role in many natural and industrial processes, and molecular simulations have proven vital in uncovering its kinetics and mechanisms. A fundamental component of such simulations is the choice of an order parameter (OP) that quantifies the progress of nucleation, with the efficacy of an OP typically measured by its ability to predict the committor probabilities. Here, we leverage a machine learning framework introduced in our earlier work [Domingues et al., J. Phys. Chem. Lett. 15, 1279, (2024)] to systematically investigate how key implementation details influence the efficacy of standard Steinhardt OPs in capturing the progress of both homogeneous and heterogeneous ice nucleation. Our analysis identifies distance and q6 cutoffs as the primary determinants of OP performance, regardless of the mode of nucleation. We also examine the impact of two popular refinement strategies, namely chain exclusion and hydration shell inclusion, on OP efficacy. We find neither strategy to exhibit a universally consistent impact. Instead, their efficacy depends strongly on the chosen distance and q6 cutoffs. Chain exclusion enhances OP efficacy when the underlying OP lacks sufficient selectivity, whereas hydration shell inclusion is beneficial for overly selective OPs. Consequently, we demonstrate that selecting optimal combinations of such cutoffs can eliminate the need for these refinement strategies altogether. These findings provide a systematic understanding of how to design and optimize OPs for accurately describing complex nucleation phenomena, offering valuable guidance for improving the predictive power of molecular simulations.