The "Money Exchange Model" is a type of agent-based simulation model used to study how wealth distribution and inequality evolve through monetary exchanges between individuals. The primary focus of this model is to identify the limiting wealth distributions that emerge at the macroscopic level, given the microscopic rules governing the exchanges among agents. In this paper, we formulate generalized versions of the immediate exchange model and the uniform saving model both of which are types of money exchange models, as discrete-time interacting particle systems and characterize their stationary distributions. Furthermore, we prove that under appropriate scaling, the asymptotic wealth distribution converges to a Gamma distribution or an exponential distribution for both models. The limiting distribution depends on the weight function that affects the probability distribution of the number of coins exchanged by each agent. In particular, our results provide a mathematically rigorous formulation and generalization of the assertions previously predicted in studies based on numerical simulations and heuristic arguments.