There is an easy way to tell if you are doing a true vibrato or not. When using your diaphram, your lower stomach should expand. Also, when doing vibrato on oboe, make sure you are vibrating your airstream not lips.
vibrato actually comes from the throat more than the diaphragm. the diaphragm is good for support, but in order to get a faster, more brilliant vibrato, you really need to learn to use your throat.
My professor taught me a good exercise to learn to use your throat. Whistle "Yankee Doodle," but very fast, and very staccato. You should here a sort of sound in your throat, and feel your throat pulsing. Its a good way to understand the concept of throat vibrato