Recommendation of actuator

First, since steppers are great at positioning (there is no need for a position feedback), you should certainly limit their movement as you've said yourself. I am not sure how the motor shaft is engineered right now, but if it was fixed to the motor, letting it continue spinning would risk damaging the equipment.

Next, 200ms transport delay in your sensor will probably be too slow, otherwise you will need to slow things down a lot in order to slow down the ball itself. Similar to what Rocket Surgeon said, you should simplify the image processing algorithm to calculate the path only once, and then quickly calculate only the position of the ball in each frame. If you want to skip this step quickly, find a red ball instead of this one, and then check only the red component in your RGB picture, until you've found a better algorithm.

For the PID control, start with the fact that you actually need two separate PID controllers, one for the east-west motor, the other one for the north-south one. If you have two exact motors, their parameters must be equal.

For a PID controller to act, it needs to know the error: difference between the desired position, and the actual position of the ball. X and Y components of this offset will be the inputs for two PID controllers (one for each motor). To get the error, you need to have the desired position on your path first: a trajectory.

To get the trajectory, you need to process the image and get the path, as well as its starting and ending point. I am not sure if your algorithm is capable of distinguishing the path from the rest of the board right now, but if not, note that this is an algorithm of its own to handle before continuing. Again, you can skip this part by manually entering the junction points, if you are eager to see some results quickly. In any case, you should be able to define the setpoint speed, and have your software move the desired coordinate position over the path, from start towards the end. Obviously, you will start with a low desired speed.

So, before starting with control, you should go through the following checklist first:

• Simplify your image processing algorithm to get faster response
• Create an algorithm which creates a trajectory over your path using a predefined speed
• In each frame:
  • Calculate the difference between the trajectory and the ball position
  • Pass the delta-X component to the east-west PID, pass the delta-Y to the north-south PID

It may turn out that it is better to create the trajectory one segment at a time, and continue with the next segment when that ball ends the previous one. Otherwise, you will need to take care that the ball doesn't overshoot the desired trajectory (which may be hard to accomplish)

I don't know what is in your CD player, but it sounds like you are asking in general how to create linear mechanical motion that is electrically controlled.

The most obvious answer is a solenoid. Look around and you will see many different types. These are basically a magnetic plunger moved by the magnetism of a coil.

There are also devices known as voice coils, which are the same concept as a solenoid except that the coil is intended to move. They are called voice coils because this is the mechanism used to make loud speaker cones move.

There are also such things as linear motors. These are like rotary motors with the magnetic poles sortof unwrapped in a line.

Of course rotary motion, such as produced by a electric motor, can be turned into linear motion mechanically. A rack and pinion arrangement is one way. There can also be lever-arm driven mechanisms, which are in turn driven by geared down motors.

Getting accuracy is a separate issue. A basic solenoid is generally intended to be on or off. Voice coils can position quite accurately as a function of current when working against a known mechanical spring force. If a stepper motor is used, then the rotary motion is known open loop, and this can be turned into known linear movement. Otherwise, you will need some kind of mechanical position sensing and closed loop feedback to control the actuator. There are again many types of that, but that's getting too far afield of your question.

Added:

Whoever downvoted this answer, please explain what you think is wrong. Phantom downvotes don't benefit anyone since nobody knows what you object to. It also doesn't give anyone a chance to decide whether you might be wrong.