Gravy is a liquid or a semisolid liquid mixture cooked along with meant or vegetables and a lot of spices. Sauce is essentially a moist or liquid component, served along with the dish to add contrast and complementary flavours. All gravies are sauces. But all sauces are not gravies. (source)
I agree with your definitions of gravies and sauces, therefore it follows that a gravy is a sauce but a sauce isn’t, necessarily, a gravy.