Cell proliferation and cell movement are fundamentally stochastic processes which lead to variability in the growth and spatial structure of cell populations in many biological settings, such as cell invasion, wound healing, and tumour growth. Understanding how variability in cell-cycle progression and local spatial constraints interact to impact population-level behaviour requires modelling frameworks that explicitly represent individual-cell behaviour and elucidate how these affect emergent tissue-level behaviour. We develop stochastic, on-lattice agent-based models (ABMs) which incorporate volume exclusion, random movement, and multi-stage representations of the cell cycle. The multi-stage framework enables a more realistic representation of true cell-cycle time distributions. We also introduce a novel form of myopic behaviour, in which cells sense their local environment when attempting to proliferate. For each ABM, we derive a corresponding continuum partial differential equation (PDE) description under the mean-field approximation. Using numerical simulations, we investigate how different proliferation mechanisms influence population-level dynamics in both the discrete and continuum models. In particular, we consider biologically relevant contexts of growth-to-confluence assays (using uniform initial conditions) and travelling wave behaviour associated with cell invasion. We examine how the PDE solutions compare with the behaviour of the corresponding ABMs averaged over many realisations.