An unified diagnostic framework is proposed to bridge theoretical, phenomenological, and process-oriented approaches for investigating the Madden-Julian Oscillation (MJO). Building upon a physical theory (moisture mode theory in this study) and linear inverse modeling, the framework links the statistical behavior of observable indices to the underlying physical processes governing column moisture evolution. Applied to the MJO in ERA5 reanalysis and 15 CMIP6 models, the framework reveals that most models simulate an MJO that propagates too slowly eastward across the basins, and decays too rapidly, especially over the Maritime Continent. By projecting model biases in column-integrated water vapor-based MJO indices onto individual terms of the moisture budget, we diagnose the physical origins of their errors. Systematic biases are primarily tied to misrepresented horizontal moisture advection and compensating errors between vertical moisture transport and convective drying, while their relative importance varies across basins. This process-resolved perspective explains the inter-model diversity in MJO simulations and provides a physically interpretable bridge between dynamical theory, model evaluation, and observational constraints—offering a transferable framework for diagnosing variability in other climate phenomena.