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.