This study aims to harmonize two frameworks commonly used to understand precipitation partitioning within a catchment: the one-stage and two-stage partitioning frameworks. The one-stage framework, exemplified by the Budyko method, partitions precipitation into runoff and evapotranspiration. The two-stage framework, partitions precipitation into infiltration and surface runoff, and further divides infiltrated water into baseflow and evapotranspiration. The second-stage of the latter has recently been recently characterized as the Horton-Index framework. In this study, we transformed Budyko equations (including Turk-Pike-1964, Fu-1981, Zhang-2001, and Wang-Tang-2014) to the Horton-Index framework. We calibrated the transformed equations, and assessed their performance at 300+ US catchments. Moreover, we propose regression models for a priori parameter estimation. We found that parameters calibrated/estimated for one framework could be reliably applied to the other, indicating the similarity of the partitioning frameworks. This lays the groundwork for unifying the two approaches, which may foster a more comprehensive understanding of hydrological processes.