It seems to me that the power company can’t tell WHEN you are using your electricity, only HOW MUCH. So, is it cheaper at night or is that an urban legend?

No, it’s true. There is less demand at night, so if the power companies can get customers to shift their usage to some time at night, they can have more capacity during the day. Demand goes up during the day because more people are active then, more people run their AC units then in the summer, and more people keep their houses warmer during the day in the winter.

Some electric meters are set up to record WHEN you use power, as well as how much. And as more plug-in electric cars come on the market, I expect that you’ll see more efforts by utilities to get people to charge them at night. Your charging system may have a timer on it to let it wait until after midnight to start charging.