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Recently, a new conceptual framework for deep convection scheme triggering and closure has been developed and implemented in the LMDZ5B general circulation model, based on the idea that deep convection is controlled by sub-cloud lifting processes. Such processes include boundary-layer thermals and evaporatively-driven cold pools (wakes), which provide an available lifting energy that is compared to the convective inhibition to trigger deep convection, and an available lifting power (ALP) at cloud base, which is used to compute the convective mass flux assuming the updraft vertical velocity at the level of free convection. While the ALP closure was shown to delay the local hour of maximum precipitation over land in better agreement with observations, it results in an underestimation of the convection intensity over the tropical ocean both in the 1D and 3D configurations of the model. The specification of the updraft vertical velocity at the level of free convection appears to be a key aspect of the closure formulation, as it is weaker over tropical ocean than over land and weaker in moist mid-latitudes than semi-arid regions. We propose a formulation making this velocity increase with the level of free convection, so that the ALP closure is adapted to various environments. Cloud-resolving model simulations of observed oceanic and continental case studies are used to evaluate the representation of lifting processes and test the assumptions at the basis of the ALP closure formulation. Results favor closures based on the lifting power of sub-grid sub-cloud processes rather than those involving quasi-equilibrium with the large-scale environment. The new version of the model including boundary-layer thermals and cold pools coupled together with the deep convection scheme via the ALP closure significantly improves the representation of various observed case studies in 1D mode. It also substantially modifies precipitation patterns in the full 3D version of the model, including seasonal means, diurnal cycle and intraseasonal variability.