I don't know what the right term for the machine is, but I would like to look into machines that can move something with two bars. For example, the machine that moves the basketball hoop in Stuff Made Here's never-miss basketball hoop. Does anyone know what this is actually called and how I might start to build one that can move, say, six inches in any direction?
The overall "study" of such movement, particularly with regard to the math, mechanics and operation/programming is called kinematics. You'll find references to x-y or cartesian kinematics and in the 3D printer world, delta kinematics. There's a few other unusual implementations such as the hang printer, a few arm printer designs with one using polar kinematics.
For light, fast x/y movement, consider to check out coreXY studies/constructions. The stepper motors are fixed to a frame, with the belts running through a series of pulleys to create carriage movement.