When you are throwing your rear hand then your weight should be on your front leg, also helps you to move backwards and pivot more easily, maybe he was talking about that? Whatever hand you are throwing the weight should be on the opposite leg, not too much though and when you are not throwing the weight should be distributed evenly. Apart from uppercuts, then your weight should be on the same side as the shot.
Leaning forward is a good way to box if you wanna have your head taken off by an uppercut that will fit snuggly on your chin/throat and i doubt youll see it coming from the posture itll leave you with.
You guys are abstracting things too much, even if what you're saying is true. If he has a real good rep just run with it, trainers work in mysterious ways. Even if it doesn't work something good will have rubbed off.