For ships yes, but for dive boats its going to depend on where you are. If you are anchored or moored 12miles offshore in say NC, jumping overboard thinking you are going to rescue someone is not a good idea. And I don't think the CG would see you as anything but underway and leaving your vessel not undercommand either. The same situation in a harbor would probably be viewed as not underway and at anchor.
The CG is great at splitting hairs about this stuff but the reality is that incidents (and a lost/drowned diver would be one) instantly end up in the court system where it all comes down to what a prudent mariner would do. If you rescued the diver in distress it would be good. If your customers had to start the boat and pick you up or radio a helicopter to come find you offshore, you can expect to lose your license.
It can be dangerous to start redefining words that already have very precise definitions in the law and regulations. For example, "not under command" does not mean "without a captain on board" as you seem to think. "Not under command" means "disabled such that the vessel is maneuvering with difficulty."