Artificial ice systems have been designed to replicate paradigmatic phenomena observed in frustrated spin systems. Here, we present a detailed theoretical analysis based on Monte Carlo simulations of the low-energy phases in an artificial colloidal ice system, a recently introduced ice system where an ensemble of repulsive colloids are two-dimensionally confined by gravity to a lattice of double wells at a one-to-one filling. Triggered by recent results obtained by Brownian dynamics simulations [A. Libál et al., Phys. Rev. Lett. 120, 027204 (2018)], we analyze the energetics and the phase transitions that occur in the honeycomb geometry (realizing the analog of a spin-ice system on a kagome lattice) when decreasing the temperature. When the particles are restricted to occupy the two minima of the potential well, we recover the same phase diagram as the dipolar spin-ice system, with a long-range-ordered chiral ground state. In contrast, when considering the particle motion and their relaxation within the traps, we observe ferromagnetic ordering at low temperature. This observation highlights the fundamental role played by the continuous motion of colloids in artificial ice systems.