What is the relevance of major mergers and interactions as triggering mechanisms for active galactic nuclei (AGNs) activity? To answer this long-standing question, we analyze 140 XMM-Newton-selected AGN host galaxies and a matched control sample of 1264 inactive galaxies over z ~ 0.3-1.0 and M * < 1011.7 M sun with high-resolution Hubble Space Telescope/Advanced Camera for Surveys imaging from the COSMOS field. The visual analysis of their morphologies by 10 independent human classifiers yields a measure of the fraction of distorted morphologies in the AGN and control samples, i.e., quantifying the signature of recent mergers which might potentially be responsible for fueling/triggering the AGN. We find that (1) the vast majority (>85%) of the AGN host galaxies do not show strong distortions and (2) there is no significant difference in the distortion fractions between active and inactive galaxies. Our findings provide the best direct evidence that, since z ~ 1, the bulk of black hole (BH) accretion has not been triggered by major galaxy mergers, therefore arguing that the alternative mechanisms, i.e., internal secular processes and minor interactions, are the leading triggers for the episodes of major BH growth. We also exclude an alternative interpretation of our results: a substantial time lag between merging and the observability of the AGN phase could wash out the most significant merging signatures, explaining the lack of enhancement of strong distortions on the AGN hosts. We show that this alternative scenario is unlikely due to (1) recent major mergers being ruled out for the majority of sources due to the high fraction of disk-hosted AGNs, (2) the lack of a significant X-ray signal in merging inactive galaxies as a signature of a potential buried AGN, and (3) the low levels of soft X-ray obscuration for AGNs hosted by interacting galaxies, in contrast to model predictions. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA Inc., under NASA contract NAS 5-26555; the XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA; European Southern Observatory under Large Program 175.A-0839; and the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.