Recording motion/movement sequence efficiently & accurately
Total re-worked answer:

The key is just make sure the data going to the motor controller is the same whether via the joystick or the memory. So as long as the data you store and the data from the joystick can be converted to an identical signal, which is then fed to the motors, it can't possibly do anything but give the same result.

One aspect that should be helpful, is given momentum of the motor, digital steps with a high enough resolution will virtually eliminate jittery camera movements. Plus you can also have circuitry to smooth out the steps.

So ultimately motors will need a signal which corresponds to a velocity. And the position of the joystick corresponds to that velocity. However, the signals between them can have an intermediate form, which could be acceleration, position, or velocity. 

You say that memory storage will be a problem. But as a minimum you will need to store data which represents the position of the joystick. There is no way around that. If you want a sample rate of 10Hz, and the joystick has a high resolution, then it's going to need roughly 10 values recorded per second. But over 60 seconds, this only needs 600 words stored in RAM, which is comfortably available in low cost processors. For example, the AT91SAM3X8E (as used on the Arduino Due) has 96kB, which at 10Hz sampling of X and Y gives 80 minutes recording time.