According to her, vibrato is produced from the nose to your lower abdomen, and it requires all of it to produce vibrato.

The fact that she believes the nose to be involved in production of vibrato is concerning. The nose is not related to flute playing unless you're using a nose flute. It must be pretty much closed off to the air so as not to result in unnecessary leakage as you're playing. As Patrick pointed out, you need your abdominal muscles to support your sound, but they are not the source of the vibrato. That comes from the throat. Trying to use your abs to produce vibrato can actually destabilize your sound, or will result in a vibrato that is so slow that it can hardly be called vibrato. Her idea that the throat should not be involved in vibrato is a long held misconception that is beginning to come to light, so the fact that she has not heard otherwise is not surprising, but it is still a misconception. As for double tonguing, the only muscle you should have to develop is your tongue. It must move differently to fit two syllables in than it would for single tonguing, but everything else should remain the same. It sounds to me like you're trying to cut the air off with the back of your throat rather than your tongue. The throat should not have anything to do with articulation.

try to think of it this way, the air in your belly is like water in a tank, the airstream is like the hose leading the water out, it moves at a constant speed and pressure, all that changes the flow at the embochure is like the nozzle on the hose, if it is a thin airstream, or flow of water, it will appear to go faster, or spin, as vibrato does, but to say one uses the stomach in vibrato is like saying the entire tank of water is turbulent...

Let me clear some things:

She did not say that the throat doesn't help at all, but it shouldn't JUST come from the throat.

And about the nose thing, there's no actuall air coming out of my nose, but it just flares with each vibrato...shake.

Let me clear some things: She did not say that the throat doesn't help at all, but it shouldn't JUST come from the throat.

I think we understood that (I did at least, and Patrick almost certanly did as well), but the throat really is where it comes from. The abdominals support the sound/airstream, and those muscles in your throat create the vibrato. Your nose is not involved, nor should your abs/diaphragm be.

And about the nose thing, there's no actuall air coming out of my nose, but it just flares with each vibrato...shake.

That may be true, and there's nothing wrong with that other than it takes more effort that you could be directing elsewhere, but whether or not your nostrils flare when you use vibrato does not dictate what it sounds like or whether or not it's present. It's most likely a mental thing....You can see the nostrils flaring, and you can hear the vibrato at the same time, so you assume that the two are related. It would be like being taught to lift your left leg whenever you move above C3...it won't do too much to hurt things, so you may assume it's necessary, but it really doesn't help at all either.

my vibrato only comes from the throat...