Is this true or not?

Do you leave your light switch on with no bulb in socket? Or off? That's essentially what leaving key in run position is doing. If radio listening is your fancy turn key 1 click back and listen away.
No bulb means no circuit. I don't think that you could say the same about a charged electrical system. If current is flowing (circuit), something is going to get hot I would think. I agree with @Uncle Bob on that one.