Optimal transport (OT) is a general framework for finding a minimum-cost transport plan, or coupling, between probability distributions, and has many applications in machine learning. A key challenge in applying OT to massive datasets is the quadratic scaling of the coupling matrix with the size of the dataset. [Forrow et al. 2019] introduced a factored coupling for the k-Wasserstein barycenter problem, which [Scetbon et al. 2021] adapted to solve the primal low-rank OT problem. We derive an alternative parameterization of the low-rank problem based on the latent coupling (LC) factorization previously introduced by [Lin et al. 2021] generalizing [Forrow et al. 2019]. The LC factorization has multiple advantages for low-rank OT including decoupling the problem into three OT problems and greater flexibility and interpretability. We leverage these advantages to derive a new algorithm Factor Relaxation with Latent Coupling (FRLC), which uses coordinate mirror descent to compute the LC factorization. FRLC handles multiple OT objectives (Wasserstein, Gromov-Wasserstein, Fused Gromov-Wasserstein), and marginal constraints (balanced, unbalanced, and semi-relaxed) with linear space complexity. We provide theoretical results on FRLC, and demonstrate superior performance on diverse applications -- including graph clustering and spatial transcriptomics -- while demonstrating its interpretability.