I live in Alabama and all the cattle farms I know of do exactly what you describe in the second list. The cows graze on grass, and in the Winter they eat hay that was grown during the Summer. Maybe when they're sold somebody will feed them grain before they're slaughtered, but I'm not even sure about that. I do know that some of the farmers sell beef directly to the consumer, and it's delicious!
The man I was working with today is one of Alabama's many small ranchers - averages around 50 head. He eats a LOT of beef, and it's all grass-fed beef.