Yeah, pretty much this. But the law in Italy defines cheese as something made directly from milk, whereas ricotta is made mostly from whey (you can add some milk, but it's secondary).

Similarly things like mascarpone and yogurt which are not fermented using rennet can't be called cheese.

Regarding the type of whey used to make ricotta it can come from any cheese. In central Italy sheep ricotta is more prized than cow, but you can easily find buffalo and goat ricotta as well.