If you pry the slot open with a screwdriver, it will remove from the post easily. Some cocoa cola might eat through the corrosion between the post and clamp.
If the post itself moves, there’s already some serious issues. Do the sides of the battery look like they are bulged out at all? They should be completely flat.
The cable hooks on the terminal with a clamp and two bolts.Is the end of your battery cable the modern thin metal style? Or do you have a clamp on old fashion lead looking one?
They do sell pullers to remove cables from batteries. It is similar to a gear puller that should be available at most auto parts stores. It has thin jaws that push in under the clamp when operated. Got mine many years ago from Canadian Tire.
The tiny movement of the battery post that you indicated should not be an issue but certainly do not go any farther.