Is there a name for a type of grid you might find in Battleship? Where coordinates don't relate to points on a grid but rather the squares themselves?
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"Grid" is as good a name as any: See Regular Grid in Wikipedia: In particular, see the "related" grid: the Cartesian Grid:
You could also refer to this sort of "playing field" in a game like battleship as an incidence matrix of sorts: where a cell in the ith row and jth column might be occupied, using "$1$", or not occupied, using "$0$".
How about alpha-numeric grid? That the coordinates refer to squares rather than vertices is no big deal. Squares can be identified with vertices in the dual of the square lattice (which is also a square lattice).