Oh, I agree!
There probably are regional definitions coming in to play here, but in my family mush was made from cornmeal: cooked, put into a buttered bread dish, chilled, sliced, fried in bacon grease for breakfast, and usually served with the Easter or Christmas ham remains.
Grits and polenta use the coarser corn grits grinds.
And Mark, obviously your granny wasn't frying the mush in bacon fat! It was wonderfully crisp on the outside, creamy on the inside, and imbued with the taste of smoked bacon. LOL