A force causes a mass to accelerate. In this video we are dealing with one force that is constant and directed downwards i.e. the gravitational force and a second variable force supplied by either a tether on a ball or by a bucket on water.
Let us start with the simplest situation, the ball on a tether moving in a horizontal circle. The gravitational force has a constant magnitude and direction (down) and the centripetal force has a constant magnitude but a constantly changing direction i.e. always towards the hand. At any instant the net force is the square root of the sum of the squares of the component forces. For the tether to be almost horizontal therefore the centripetal force must be much larger than the gravitational force.
Now get the ball going in a vertical circle. Gravity is constantly pulling down but to get a constant angular velocity for the ball the net force has to have a constant magnitude but constantly changing direction (towards the hand) so at any instant in time the required centripetal force less the gravitational force (vector subtraction) gives us the tension force required in the tether. At the bottom this means the tension is a maximum and at the top it is a minimum. If the ball is going slow enough the tension can even be zero at the top.
The bucket of water is a bit more complex. Gravity is constantly pulling the water down. At the bottom of a vertical circle it is the bucket that is acting against gravity to hold the water and causing the water to have centripetal motion. So there is are action reaction forces between the water and the bucket. Again the tension in the handle of the bucket and your arm is at a maximum at the bottom and at a minimum the top for the same reason i.e. the tension is the required centripetal force less the gravitational force (vector subtraction). Again if you are swinging the bucket slow enough you will reach a critical situation where there is no tension in your arm and the water just stays in the bucket without any reaction force between the water and the base of the bucket. If the bucket moves any slower than this near the top of the swing the gravitational force is greater than the required centripetal force so the water follows a path determined by gravity instead of the bucket i.e. it has an initial velocity in the direction of motion and a downward acceleration producing a well recognized parabolic trajectory which may indeed take it outside of the bucket which is on a circular path.