Tiger is consistent in describing his keys for his swing. He has said in many interviews that he feels the swing in his left hand, and when he is in a tournament he feels like that awareness gets heightened.
His comment of about working on different body positions but then going back to his hands is also revealing. It's key to know what you need to do to play well, but also what you need to do to support playing well. For instance, sometimes you might want to work on an area of your swing that you don't have as much control over (for some it's the legs, or core, or hands). But then it's important to find a way to relate it to an area that you do have control over. In Tiger's case, while he might practice his body, eventually, he has to connect that body feel to a hand key in order to use it on the course. Great insight!
As for the wedge video, he's talking about turning it down to take off-spin, he's demonstrating more supination, which would support more shaft lean and higher face contact. That higher face contact would take off-spin. When he's describing adding spin and lengthening it out, he's likely talking about more ulnar deviation and more chop action to take the hand path left.