Actually, roller chain is zero backlash as long as sprockets are tight to the shafts and chains are slightly tensioned.  Belts, unless use as a primary (motor to final drive), have too much flex.  You can have backlash just from belts stretching.  The commercial plasma machines that use belt drive use something that looks like a Harley drive belt.  With the holes he posted a photo of, the backlash should be easily detected by tugging on the axes with the machine turned on.
We just serviced a router with rack and pinion that had the same issue.  A set screw had come loose on the Y axis pinion.  You could feel the backlash by tugging on the Y axis.
Looks like a router.  Ball screw would be the best option.  Ball screws are hard to use on plasma machines because of the cutting speeds needed.