You beat me to it. Most use the terms "cross" and "straight" in boxing as if it is the same punch and it is not. They are mechanically different. The cross as more of an "arc" to it because it is going over the opponent's punch.
As to the "reverse punch" chambering at the waist and a boxer doesn't. At some point, you should be punching from where your hands are in karate and not "chambering" your hand for no reason. The chambering is a training tool and does have some application for grabbing/holding and pulling the opponent into a punch. But, you should have your hands up no matter what.