A good vibrato starts in the elbow, moves the forearm, which move the hand through the wrist hinge, which is transferred as the hands for the fingers to control.
Look in the two videos again and look at how the elbow is used.....
Learn to move the elbow independent of the fingers and let it create the vibrato, rather than your fingers creating vibrato.
There are certain situations where you can and will use your fingers for a lighter vibrato, or even a wider vibrato to your notes timbre.
Again it needs practice and it needs development, so target the elbow movement first, see it as the 'rudder' that creates the vibrato and the fingers as controlling it.