Owing to both electronic and dielectric confinement effects, two-dimensional organic-inorganic hybrid perovskites sustain strongly bound excitons at room temperature. The intrinsic optical lineshape reflects multiple excitons with distinct binding energies, each dressed differently by the hybrid lattice. Given this complexity, a fundamentally far-reaching issue is how Coulomb-mediated many-body interactions --- elastic scattering such as excitation-induced dephasing, inelastic exciton bimolecular scattering, and multi-exciton binding --- depend upon the specific exciton-lattice coupling. We report the intrinsic and density-dependent exciton pure dephasing rates and their dependence on temperature by means of a coherent nonlinear spectroscopy. We find exceptionally strong screening effects on multi-exciton scattering relative to other two-dimensional single-atomic-layer semiconductors. Importantly, the exciton-density dependence of the dephasing rates is markedly different for distinct excitons. These findings establish the consequences of particular lattice dressing on exciton many-body quantum dynamics, which critically define fundamental optical properties that underpin photonics and quantum optoelectronics in relevant exciton density regimes.