Methods to accurately move an object across a 2-D plane
You might as well combine the measurement and replication into one system. Let me start with my previous answer to your question: Idea 4

You already have linear bearings and encoders set up to measure the motion of the object. Now all you need to do is to actuate those bearings. Normally, if you were building something like a CNC milling machine, and you wanted to actuate linear motions, you'd use a ballscrew and a stepper motor.



The problem with these is that they are not back-drivable, and therefore you won't be able to move the object yourself. Two options:


Add force sensors (using strain gauges). Now the system can detect you pushing on the object, and it will drive the motor to create the illusion that there's no motor there at all. This can work extremely well if you do it carefully, and if the motor is fast enough. This is known as active compliance.
Use linear motors. These are completely back-drivable when unpowered.






The good news is that, if you can afford it, you can buy complete linear motor motion systems from companies like Aerotech or Baldor Motion. Alternatively, you can ask a small robotic engineering company to build you the whole system. Again, these systems are very accurate. They're used where 0.01mm accuracy is a common requirement, and even 0.001mm accuracy (but then you really need a temperature controlled environment).