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Context.Recent JWST observations have measured the ice chemical composition towards two highly extinguished background stars, NIR38 and J110621, in the Chamaeleon I molecular cloud. The observed excess of extinction on the long-wavelength side of the H₂O ice band at 3 μm has been attributed to a mixture of CH₃OH with ammonia hydrates NH₃·H₂O), which suggests that CH₃OH ice in this cloud could have formed in a water-rich environment with little CO depletion. Laboratory experiments and quantum chemical calculations suggest that CH₃OH could form via the grain surface reactions CH₃+ OH and/or C + H₂O in water-rich ices. However, no dedicated chemical modelling has been carried out thus far to test their efficiency. In addition, it remains unexplored how the efficiencies of the proposed mechanisms depend on the astrochemical code employed. Aims.We modelled the ice chemistry in the Chamaeleon I cloud to establish the dominant formation processes of CH₃OH, CO, CO₂, and of the hydrides CH₄and NH₃(in addition to H₂O). By using a set of state-of-the-art astrochemical codes (MAGICKAL, MONACO, Nautilus, UCLCHEM, and KMC simulations), we can test the effects of the different code architectures (rate equation vs. stochastic codes) and of the assumed ice chemistry (diffusive vs. non-diffusive). Methods.We consider a grid of models with different gas densities, dust temperatures, visual extinctions, and cloud-collapse length scales. In addition to the successive hydrogenation of CO, the codes’ chemical networks have been augmented to include the alternative processes for CH₃OH ice formation in water-rich environments (i.e. the reactions CH₃+ OH → CH₃OH and C + H₂O → H₂CO). Results.Our models show that the JWST ice observations are better reproduced for gas densities ≥10⁵cm⁻³and collapse timescales ≥10⁵yr. CH₃OH ice formation occurs predominantly (>99%) via CO hydrogenation. The contribution of reactions CH₃+ OH and C + H₂O is negligible. The CO₂ice may form either via CO + OH or CO + O depending on the code. However, KMC simulations reveal that both mechanisms are efficient despite the low rate of the CO + O surface reaction. CH₄is largely underproduced for all codes except for UCLCHEM, for which a higher amount of atomic C is available during the translucent cloud phase of the models. Large differences in the predicted abundances are found at very low dust temperatures (T_dust<12 K) between diffusive and non-diffusive chemistry codes. This is due to the fact that non-diffusive chemistry takes over diffusive chemistry at such low T_dust. This could explain the rather constant ice chemical composition found in Chamaeleon I and other dense cores despite the different visual extinctions probed.