Here's what I was taught and what I taught my Engineering students:
Use the "Right Hand Rule": put your thumb and forefinger on your right hand out like you are making a gun. They will be at right angles to each other in the same plane as your palm. Stick your middle finger out at right angles to your palm ("normal" to your palm, if you prefer that term; it's the same). Your thumb is the X-axis, your pointer finger is the Y-axis and your middle finger is the Z-axis, all pointing in the positive direction. The Z axis is the axis of rotation on any machine.
So a lathe technically doesn't have an X-axis, just Y and Z.
We tend to refer to the left and right direction as the X-axis, and forward and back as the Y and up and down as the Z. Using the RHR gives us a common language and reference for any machine.