Understanding the magnetization dynamics induced by spin transfer torques in perpendicularly magnetized magnetic tunnel junction nanopillars and its dependence on material parameters is critical to optimizing device performance. Here we present a micromagnetic study of spin-torque switching in a disk-shaped element as a function of the free layers exchange constant and disk diameter. The switching is shown to generally occur by 1) growth of the magnetization precession amplitude in the element center; 2) an instability in which the reversing region moves to the disk edge, forming a magnetic domain wall; and 3) the motion of the domain wall across the element. For large diameters and small exchange, step 1 leads to a droplet with a fully reversed core that experiences a drift instability (step 2). While in the opposite case (small diameters and large exchange), the central region of the disk is not fully reversed before step 2 occurs. The origin of the micromagnetic structure is shown to be the disks non-uniform demagnetization field. Faster, more coherence and energy efficient switching occur with larger exchange and smaller disk diameters, showing routes to increase device performance.