Convective rainfall varies at multiple scales: individual cells evolve within storms, while the population of cell intensities can differ substantially between storms. Existing object-based stochastic storm generators commonly represent cell occurrence and movement but provide limited representation of these two sources of variability. This study develops a spatial-temporal convective storm generator that explicitly represents both. At the cell scale, radar-derived tracks are represented using three lifespan-based classes, ranging from complete growth-peak-decay lifecycles to short-lived and transient cells, with dependence among evolving intensity and geometric properties preserved using copulas. At the storm scale, each event is assigned a storm-specific distribution of cell intensities. Its location and spread are modelled using Generalised Additive Models for Location, Scale and Shape (GAMLSS), with near-surface air temperature representing systematic environmental dependence and event-level random effects representing residual storm-to-storm variability. The framework was calibrated using 169 warm-season convective storms observed over Birmingham, UK, between 2005 and 2017. The resulting generator reproduces the observed cell-property distributions, dependence structures, storm-scale organisation and temporal clustering. Importantly, using a single pooled cell-intensity distribution largely removes the observed between-storm intensity variability, whereas the storm-specific formulation recovers the observed variance partition while retaining within-storm dispersion. This improves the distributions of event-maximum cell intensity and substantially improves their extreme-value behaviour, with observed return levels remaining within the 5-95% simulation intervals across return periods of 2-100 years. The resulting framework links storm-scale environmental variability to populations of dynamically evolving convective cells, providing a computationally efficient basis for large-ensemble spatial-temporal rainfall generation and future climate-conditioned applications.