I was taught the same thing-- to allow incoming things (swords) to be deflected off to the side away from you. In other words some measure of passive protection is built into the posture, while any active defense would be on top of that. (Like a boxer's hands being up-- but since the sword is thin, it would only guard one sliver of your body if not tilted)
In the Iwama system the "tilt" is a manifestation of the hanmi stance with all it entails. In the start of the second video, for example, it is there on the finishing cut preventing the ai-uchi (mutual kill) situation.