The area of effect is under spells in the PHB, in the diagram the source is the "*". So the origin of the cube is always one side of the cube, not the center. For the caster to be at the center it would have to a sphere centered on the caster.
Cube
You select a cube's point of origin, which lies anywhere on a face of the cubic effect. The cube's size is expressed as the length of each side.
A cube's point of origin is not included in the cube's area of effect, unless you decide otherwise.
Feel free to run it any way you want of course, but the rules are quite clear.