The kind of section you made in the second picture, plane 110, is used to compute distances between the atoms. Is does not matter that the plane is not stoechiometric in that regard because you only want to know distances. But you cannot call the obtained graph a unit cell.
The unit cell of a 3D crystal should be 3D because you should be able to recreate the entire crystal from space tight duplication and translation of the unit cell. The unit cell should also be the smallest possible in volume / number of contained atoms.
In case there is another shape that also permits the recreation of the crystal by duplication / translation, but holds an easier to understand shape, then you can do the same calculations as you would with this new cell. This is the case where I would use the term "simplified cell".
Example with NaCl:
The unit cell of NaCl -shown in black- is impractical to use and understand.
Note that the usage of a simplified cell, even if it has more than the minimum possible number of atoms for a unit cell, by geometrically sharing the contribution of atoms to the current cell, does not break stoechiometry, in the NaCl example:
n(Na) = 8 * 1/8 (corners) + 6 * 1/2 (face centers) = 4
n(Cl) = 12 * 1/4 + 1 = 4