Obviously, you can define the coordinate system any way you wish. However, you'll eliminate one more potential entanglemnent (ie, source of error) if you stick with a right-handed system since that's the standard in all of mathematics.
It seems more natural to me if the +Z axis on the mill points down (the working direction of the tool), hence my choice of X and Y above.
Similarly on the lathe. As on the mill, the chuck spins CCW, so the spin vector points out of the chuck. Pick +X and +Y to preserve this relationship. I would be inclined to make +X away from the operator (ie, the way you put cut on) and +Y up.
For Zee et al...
The right hand rule. Using your right hand, orient the extended index finger along +X. Extend the middle finger perpendicular to the index and align it along +Y. Now, the extended thumb, perpendicular to the XY plane, will point in the +Z direction of a right-handed coordinate system.
In mathematical terms, +Z is the vector cross product of +X into +Y.
