difference between machine coordinate systems and CAD software

Your fist picture = you are looking at the part from behind, any particular reason for it?
Moving table and moving head are two different types of machines, despite being the same functionally, but very different when looking at them and using them, i have a feeling you are mixing these.

Machine coordinates are always defined with respect to tool movement on the workpiece regardless if it's the spindle or the workpiece that moves. So presuming that the blue coordinate system represents the tool movement on the workpiece then that is a common setup for a mill. Having the X axis pointing to the left in the CAD system is not wrong (you just have to rotate the part on the machine table) but a bit unusual.