Animal cells actively generate contractile stress in the actin cortex, a thin actin network beneath the cell membrane, to facilitate shape changes during processes like cytokinesis and motility. On the microscopic scale, this stress is generated by myosin molecular motors, which bind to actin cytoskeletal filaments and use chemical energy to exert pulling forces. To decipher the physical basis for the regulation of cell shape changes, here, we use a cell-like system with a cortex anchored to the outside or inside of a liposome membrane. This system enables us to dissect the interplay between motor pulling forces, cortex-membrane anchoring, and network connectivity. We show that cortices on the outside of liposomes either spontaneously rupture and relax built-up mechanical stress by peeling away around the liposome or actively compress and crush the liposome. The decision between peeling and crushing depends on the cortical tension determined by the amount of motors and also on the connectivity of the cortex and its attachment to the membrane. Membrane anchoring strongly affects the morphology of cortex contraction inside liposomes: cortices contract inward when weakly attached, whereas they contract toward the membrane when strongly attached. We propose a physical model based on a balance of active tension and mechanical resistance to rupture. Our findings show how membrane attachment and network connectivity are able to regulate actin cortex remodeling and membrane-shape changes for cell polarization.