No mechanisms have hitherto been conclusively demonstrated to be responsible for initiating optically-luminous nuclear (Seyfert) activity in local disk galaxies. Only a small minority of such galaxies are visibly disturbed in optical starlight, with the observed disturbances being at best marginally stronger than those found in matched samples of inactive galaxies. Here, we report the first systematic study of an optically-selected sample of twenty-three active galaxies in atomic hydrogen (HI) gas, which is the most sensitive and enduring tracer known of tidal interactions. Eighteen of these galaxies are (generally) classified as Seyferts, with over half (and perhaps all) having [OIII] luminosities within two orders of magnitude of Quasi-Stellar Objects. Only ~28% of these Seyfert galaxies are visibly disturbed in optical DSS2 images. By contrast, ~94% of the same galaxies are disturbed in HI, in nearly all cases not just spatially but also kinematically on galactic (>~20 kpc) scales. In at least ~67% and perhaps up to ~94% of cases, the observed HI disturbances can be traced to tidal interactions with neighboring galaxies detected also in HI. The majority of these neighboring galaxies have projected separations of <~ 100 kpc and differ in radial velocities by <~100 km/s from their respective Seyfert galaxies, and many have optical luminosities ranging from the Small to Large Magellanic Clouds. In a companion paper, we show that only ~15% of a matched control sample of inactive galaxies display comparable HI disturbances. Our results suggest that: i) most Seyfert galaxies (with high nuclear luminosities) have experienced tidal interactions in the recent past; ii) in most cases, these tidal interactions are responsible for initiating events that lead to their nuclear activity.