A tale of two vibratos:
In simplistic terms there are two main schools of vibrato, and most cellists combine them, to a certain extent, and mix between registers of the instrument and for differing musical effects. That said, I think most teachers and advanced cellists conceive of vibrato in a way similar to the first two instructional videos below, and the motion demonstrated in the third video (at 19:04) is a bit of an “old-fashioned” concept. If you watch closely, later in the third video, this cellist (who is a fantastic musician and player, by the way) starts to morph into a motion similar to the one demonstrated in the first two videos. And, by the same token, many great players who conceive of vibrato in the manner of the first two videos introduce a little of the rocking motion shown in the third video in the heat of the musical moment to introduce a little more passion or width as needed.
I sometimes call the two motions the “Rock and Roll” motions of vibrato, with the first two videos demonstrating “roll” of the flexible finger joints and finger tip on the string, and the third video demonstrating “rock” of the hand and finger.
There is another aspect that I’ve only seen spoken about in a video produced by an amateur, but it is an important part of vibrato she called “cling.” It is the way the finger tip “clings” to the string, allowing the finger joints to flex in a relaxed way and to alter the pitch in a cleanly focused oscillation without slipping and sliding in a random way on the string. In the dry winter months, sometimes a little moisturizer or using an emory board to file down an overly thick and hard callous can help “cling.”
No two vibratos are exactly the same — it is one of the most personal and expressive aspects of cello playing!