Grits and Mush are not the same thing, although they are both made from corn.
Grits are ground hominy, and hominy is corn treated with lye to remove the outer hulls. Hominy may be boiled, or it may be ground into grits to reduce the cooking time. Originally, the term referred to any very coarsely ground grain. Corn grits are one of the original fast foods. (Whole corn meal has a lot more food value than hominy or grits.)
Hasty Pudding is made from meal made from ground, whole corn, and boiled in water or milk, although some early cook books have it being made with other flours in addition to corn meal. Another term for this is Indian Meal Mush, or simply Mush, though in some places "Mush" may be made from other cereals such as oatmeal. When corn meal mush is made very thick, and then baked from the heat of the fire it becomes a flatbread that is called Johnny Cake or it may be called Hoe Cake. If you substitute ground oats instead of corn meal, and you bake the hot batter into a flatbread, it's called bannock.
When the mush is allowed to sit and congeal and cool, it may be then used to make Fried Mush. In the 20th century when frying of congealed mush became a common method, the distinction between mush and fried mush was dropped in many communities, and the word "mush" was applied to the fried dish.
Gruel may be made from any cereal, including Indian Corn meal, but it is not made very thick as compared to Hasty Pudding/Mush. It is more commonly a sort of thin, oatmeal soup.
LD