 Solution

A unit cell is a repeat region whose lattice translations tile the plane. A primitive unit cell represents exactly one lattice point and has the minimum repeat area for a lattice basis.

Cell A is nonprimitive; cell B is primitive. A represents two points and has area $4a^2$. B has four shared corner points and no additional interior point, represents one point and has area $2a^2$. Fractional corner counting refers to the tiling, not to fractional physical atoms.