Latent human immunodeficiency virus (HIV) infections occur when the virus occupies a transcriptionally silent but reversible state, presenting a major obstacle to cure. There is experimental evidence that random fluctuations in gene expression, when coupled to the strong positive feedback encoded by the HIV genetic circuit, act as a 'molecular switch' controlling cell fate, i.e., viral replication versus latency. Here, we implemented a stochastic computational modeling approach to explore how different promoter activation mechanisms in the presence of positive feedback would affect noise-driven activation from latency. We modeled the HIV promoter as existing in one, two, or three states that are representative of increasingly complex mechanisms of promoter repression underlying latency. We demonstrate that two-state and three-state models are associated with greater variability in noisy activation behaviors, and we find that Fano factor (defined as variance over mean) proves to be a useful noise metric to compare variability across model structures and parameter values. Finally, we show how three-state promoter models can be used to qualitatively describe complex reactivation phenotypes in response to therapeutic perturbations that we observe experimentally. Ultimately, our analysis suggests that multi-state models more accurately reflect observed heterogeneous reactivation and may be better suited to evaluate how noise affects viral clearance.