I think the confusion here is that there are many different kinds of vibrato, with playing techniques that differ from instrument to instrument and a style and nature that relates both to the instrument and the style of music you are playing.
In general, the "classical voice" has a wide unconstrained vibrato, while the "jazz/pop" voice has a straight tone over which a more constrained, tighter vibrato is layered for musical effect.
Your classical voice teachers are going to expect a wide vibrato that comes from the overall timbre of your voice -- not something that you think about delaying or constraining. The rationale for this is that if you are actively controlling your vibrato, you are compromising on your breath support or your tone in order to exert that control (or causing tension somewhere you don't want it).
The fact that Matt Smith calls a particular style of guitar vibrato "opera style" is pretty irrelevant to this issue, because he's not an opera singer--that's just what he likes to call a guitar technique that reminds him of opera. As your classical teachers have shown you, real opera singers are pretty much always in the "on" position (at least for the classical repertoire).
It sounds like you gravitate toward more of a jazz voice style of vibrato, where you establish the pitch first and then apply vibrato in a musical way.
It's important to note that "classical voice" is a very specific type of singing, that comes from a need for the human voice to compete with an entire orchestra. When doing choral singing, I greatly prefer a lighter tone and almost no vibrato, especially for harmonically complex music. But then again, if you are singing an opera chorus, I would expect to hear a great amount of vibrato from everybody. Lots of new/contemporary music requires light and nimble straight tone voices because of the harmonic and rhythmic complexity--trying to sing new music with a "classical" voice would be like trying to navigate a freight train through a minefield.
So basically, you should learn how to execute all types of vibrato, and also know when to use one style of singing vs. another.
You mentioned lung vibrato on wind instruments -- this is an option, but the "standard" technique will often vary from instrument to instrument. To my recollection, flute and oboe are the only instruments I would recommend using lung vibrato as a primary technique. As a trombonist, I will most often use jaw vibrato in both classical and jazz contexts, and occasionally add slide vibrato as another option for jazz playing. Jaw vibrato would also be used on other brass instruments, and on the saxophone (particularly for jazz) and bassoon.
Different instruments also apply classical vibrato differently. Oboe and flute will generally vibrate at the beginning of the note when playing loudly, whereas brass instruments will establish (very) loud notes with constant pitch, and then add vibrato as they decrescendo--or when playing soft and lyrically, use vibrato everywhere. (Interestingly, clarinets and french horns are almost never expected to vibrate in common technique.) Sometimes, styles of vibrato use may even vary from a "European school" to an "American school."
So, to try to draw some generalizations here, one of the main functional purposes of classical vibrato is to make something heard that would otherwise not be. Naturally soft woodwind instruments playing forte, or the human voice competing with an orchestra, need to be as loud as possible, so wide and fast vibrato is applied at the beginning of the note. At lower dynamic levels, vibrato can be reigned in and used differently. Naturally loud instruments like the trombone and trumpet compete with an orchestra best -without- vibrato, so they too use it differently. And lastly, all bets are off for contemporary or pop music.