All of the brass teachers I know (including myself) teach the jaw method.
I find it preferable to the other two methods because:
- Diaphragmatic vibrato is going to disturb your support and airstream.
- Moving the fingers back and forth is just smushing the mouthpiece against your face, which runs the risk of fatiguing your embouchure earlier or letting air escape.
In contrast, jaw vibrato makes use of a simple brain->muscle->result link, which is easy to control and shouldn't mess with your air or embouchure too much.
The only instruments that make use of a diaphragmatic vibrato are the flute and double reeds. Proper vocal technique does NOT use a diaphragmatic vibrato. (The profession is not unanimous on this so it's an annoying point of contention, but I will briefly cite two professional voice teachers and my own.) Clarinet traditionally does not use vibrato, and saxophone typically uses jaw vibrato. The reason for each of these has to do with how tone is generated and how much resistance exists in the instrument.
Flute has essentially no resistance, making diaphragmatic vibrato the only option that will not mess with your embouchure. In contrast, the double reeds have a TON of resistance, requiring a firm and static embouchure to maintain a good tone. Saxophone, similar to the brass instruments, is pretty free blowing and utilizes an embouchure that applies tension to a vibrating element. In the saxophone's case, that's the reed; in the brass family's case, it's the lips.
Jaw vibrato in both cases is not a "pinching off" of your embouchure (in brass instruments we never vibrate sharp in pitch). When you lower the jaw, you are loosening the tension and thus allowing the pitch to sag. When teaching this, (and only after a student has developed a good and consistent tone with a relaxed throat) we start very slowly, with "square wave" oscillations at about q=60. Gradually, that is smoothed out to a sine wave and sped up at different rhythmic intervals. The result is a very precise control directly to the vibrating element that does not mess with your tone.
Diaphragmatic vibrato disturbs your support because your diaphragm is your support. From elementary school through college brass teachers repeatedly insist "More air!" because that's the foundation of good tone and volume. Vibrating with your diaphragm is literally varying the air pressure behind your lips. Additionally, the diaphragm is a huge muscle compared to the jaw, making jaw vibrato a much more efficient technique to develop.