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 CH4 and 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 ≥105 cm−3 and collapse timescales ≥105 yr. CH3OH ice formation occurs predominantly (>99%) via CO hydrogenation. The contribution of reactions CH3 + OH and C + H2O is negligible. The CO2 ice 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. CH4 is 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 (Tdust<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 Tdust. This could explain the rather constant ice chemical composition found in Chamaeleon I and other dense cores despite the different visual extinctions probed.