The relative importance of atomic defects and electron transfer in explaining conductivity at the crystalline LaAlO3/SrTiO3 interface has been a topic of debate. Metallic interfaces with similar electronic properties produced by amorphous oxide overlayers on SrTiO3 Y. Chen et al., Nano Lett. 11, 3774 (2011); S. W. Lee et al., Nano Lett. 12, 4775 (2012) have called in question the original polarization catastrophe model N. Nakagawa et al., Nature Mater. 5, 204 (2006). We resolve the issue by a comprehensive comparison of (100)-oriented SrTiO3 substrates with crystalline and amorphous overlayers of LaAlO3 of different thicknesses prepared under different oxygen pressures. For both types of overlayers, there is a critical thickness for the appearance of conductivity, but its value is always 4 unit cells (around 1.6 nm) for the oxygen-annealed crystalline case, whereas in the amorphous case, the critical thickness could be varied in the range 0.5 to 6 nm according to the deposition conditions. Subsequent ion milling of the overlayer restores the insulating state for the oxygen-annealed crystalline heterostructures but not for the amorphous ones. Oxygen post-annealing removes the oxygen vacancies, and the interfaces become insulating in the amorphous case. However, the interfaces with a crystalline overlayer remain conducting with reduced carrier density. These results demonstrate that oxygen vacancies are the dominant source of mobile carriers when the LaAlO3 overlayer is amorphous, while both oxygen vacancies and polarization catastrophe contribute to the interface conductivity in unannealed crystalline LaAlO3/SrTiO3 heterostructures, and the polarization catastrophe alone accounts for the conductivity in oxygen-annealed crystalline LaAlO3/SrTiO3 heterostructures. Furthermore, we find that the crystallinity of the LaAlO3 layer is crucial for the polarization catastrophe mechanism in the case of crystalline LaAlO3 overlayers.