We analyze a suite of 33 cosmological simulations of the evolution of Milky-Way-mass galaxies in low-density environments. Our sample spans a broad range of Hubble types at z = 0, from nearly bulgeless disks to bulge-dominated galaxies. Despite the fact that a large fraction of the bulge is typically in place by z ~ 1, we find no significant correlation between the morphology at z = 1 and at z = 0. The z = 1 progenitors of disk galaxies span a range of morphologies, including smooth disks, unstable disks, interacting galaxies, and bulge-dominated systems. By z ~ 0.5, spiral arms and bars are largely in place and the progenitor morphology is correlated with the final morphology. We next focus on late-type galaxies with a bulge-to-total ratio (B/T) <0.3 at z = 0. These show a correlation between B/T at z = 0 and the mass ratio of the largest merger at z < 2, as well as with the gas accretion rate at z > 1. We find that the galaxies with the lowest B/T tend to have a quiet baryon input history, with no major mergers at z < 2, and with a low and constant gas accretion rate that keeps a stable angular-momentum direction. More violent merger or gas accretion histories lead to galaxies with more prominent bulges. Most disk galaxies have a bulge Sérsic index n ≤ 2. The galaxies with the highest bulge Sérsic index tend to have histories of intense gas accretion and disk instability rather than active mergers.