In each model, we vary one parameter and fix the other five parameters. The values of these parameters for all six models are provided in Table . We note that by changing the parameters in models 1 to 3, the polariton relaxation dynamics are affected through the system contribution to the Hamiltonian, while in changing the parameters in models 4 to 6, the dynamics are affected through the system-bath interaction term. Furthermore, the results for models 1 to 4 are presented in the main text below, while models 5 and 6 are provided in Sec. III of the Supplementary Material.