The continuity equation says that IF VOLUMETRIC FLOW IS FIXED, e.g. "always XYZ liters per second", then a smaller pipe or smaller hole causes higher fluid velocity.
The water faucet is not like that. The flow is not fixed. When the faucet is slightly open it is "X liters per second" flowing out ... when the faucet is fully open it is "Y liters per second" flowing out. Y is bigger than X. It's NOT a constant fixed flow rate. Therefore the continuity equation does NOT imply that a narrower opening means higher fluid velocity.
In the video, by the looks of it, the tube is overall hanging down. Even when wriggling the general orientation is still down.
I don't think the vena contracta aspect makes any difference here.
How I would approach this:
I imagine the water flow initially very slow, and the rate of flow is gradually increased, to explore the entire range of rates of flow.
The lower the flow rate the lower the exit velocity of the water.
When water exits a nozzle (more generally, when any fluid/gas exits a nozzle) there is a recoil force. If the nozzle would be secured to some cart then the recoil force would propel the cart.
At low flow rate the force of gravity is larger than the recoil force at the nozzle. The larger the flow rate, the larger the recoil force. At some flow rate the recoil force will be enough to lift the nozzle against gravity.
The tube is flexible, so the tube bends. That makes the nozzle point sideways. With the nozzle pointing sideways the tube is moved sideways, and gravity has opportunity once again to pull the nozzle down.
The result is that the nozzle is violently swinging from side to side, flexing the tube all the time.
In addition there is the inertia of the water flow in the tube itself. When the water is forced to move along a bend in the tube the inertia of the water will tend to straighten the bend, contributing to the overall chaos of the motion.
Best Answer
This diagram shows the difference between closing the tap and pinchng the end of the hose:
In both cases you are reducing the area the water has to flow through, and this increases the water velocity in the constriction. The upper diagram shows what happens when you close the tap. Closing the tap increases the velocity of the water at the constriction, but as soon as the water is past the constriction is slows down again and it emerges from the end of the hosepipe with a relatively low velocity.
The lower diagram shows what happens when you pinch the end of the pipe. The constriction increases the velocity of the water but because the constriction is right at the end the water doesn't have a chance to slow down again so it leaves the end of the pipe with a relatively high velocity.