Alveolar epithelial type 2 cells (AEC2s) are the facultative progenitors responsible for maintaining lung alveoli throughout life but are difficult to isolate from patients. Here, we engineer AEC2s from human pluripotent stem cells (PSCs) in vitro and use time-series single-cell RNA sequencing with lentiviral barcoding to profile the kinetics of their differentiation in comparison to primary fetal and adult AEC2 benchmarks. We observe bifurcating cell-fate trajectories as primordial lung progenitors differentiate in vitro, with some progeny reaching their AEC2 fate target, while others diverge to alternative non-lung endodermal fates. We develop a Continuous State Hidden Markov model to identify the timing and type of signals, such as overexuberant Wnt responses, that induce some early multipotent NKX2-1+ progenitors to lose lung fate. Finally, we find that this initial developmental plasticity is regulatable and subsides over time, ultimately resulting in PSC-derived AEC2s that exhibit a stable phenotype and nearly limitless self-renewal capacity. Keywords: Alveolar epithelial type 2 cells; DNA barcoding; cell fate; cellular plasticity; directed differentiation; heterogeneity; hidden Markov model; induced pluripotent stem cells; lungprogenitors; single cell RNA sequencing.