By warm up he's talking about the pre-heat system that helps diesels start when they are cold. Diesel engines ignite fuel by compression heat alone, there are no spark plugs. When it's cold, they can be harder to start. Two approaches, one is to heat the intake air with a heating element in the intake, the other is to have a glow plug (heating element) in each cylinder. Once the engine is running those pre-heating devices shut off.
With a diesel car, it's better to not let them "warm up" after starting... diesels don't warm up very quickly if they are not under load and they will accumulate carbon and parts will wear more if you let them "warm up" at idle after starting.
I'm not saying this is the only reason for the lack of warm up (especially today), but it was a significant part of the perceived reduction from old diesels. It's also why if you put petrol in the car by mistake, you can still ruin the engine even before you've turned the ignition (as opening the door starts the warm up and fuel pumps etc)
Modern car aren't actually completely on or off when you turn the key, sure. However I've put petrol by mistake in a diesel car twice and didn't ruin anything (but lost some money cleaning the mess up).
