When you puncture the container I don't understand why the velocity of the water have vertical velocity. The water is falling due to the pressure of the atmosphere and the gravity.
The water will not fall in a parabola like a cannon projectile. The water will not go up and come down because this is a violation of the conservation of energy as the work will be done without the external force to do work on it.
The equation of the velocity is for the velocity of the coming water irrespective of the orientation of the opening on the container which is on the side.
It is important to remember that the velocity of the water coming out of the hole which is on the side of the container will reduce as the height of the water level will be reducing instantaneously.
But when the water is falling through the hole which is in the bottom of the container. the water will come out with constant speed. Because the velocity of the water is not dependent on the water column.
Try out this experiment using a water bottle. you will get it.