Electronic – Small cheap electronic solenoid valve

A solenoid is certainly the cheapest and easiest thing you can make yourself. Plus, it's not difficult to direct the flow of the water this way. You don't even need to make a valve. All you need to do is to re-direct the flow of the water.

Since you have a flexible tube, you simply need to move the end of the tube between two positions. In one position it squirts the water down, and in the other position, it squirts the water into a drain.

You can wind the solenoid coils yourself. Just cut lengths of steel rod, and wind coil wire (enamelled copper wire around them).

Push the end of your water tube into a short length of steel tube. The springyness of the water tube will hold it away from the solenoid, and allow water to flow into the drain (This is also the fail safe position. When the power fails, it will drain). When the solenoid is energised, it will attract the tube, and allow the water to flow down and be seen.

You may find that the flexible tube doesn't actually work well as a spring. In that case add a spring. Or, use a fairly long piece of thin steel tube instead of the flexible tube. This will be flexible enough to move the approx 4mm required.

How about this. (apologies for rough sketch)

The tube passes through the end of the clamp which is shaped like a J to ensure it can't fall out when being squeezed. A round nosed piece of plastic with a screw thread through it can be moved between two guides that only allow forward and backward motion. As the FIXED (12V) motor turns the threaded screw attached to its shaft the pinch block moves forward or backwards depending on rotation direction. The adjustable limit switches are there to prevent the pinch block from moving to far in either direction.