Your Support System
I have a question to ask. Could you tell me what the real "support" is? One of my teachers said no matter what register you play, you must keep pushing your belly outward. The other teacher told me exactly the opposite thing: you have to keep pushing your belly inward, it is also called "wedge" breathing. There is also a teacher told me the real support is air speed. Could you give me some advice about that?
Thanks for your question. It seems to me that the schools of thought on this are very different, especially from country to country. I know that you have had some very qualified teachers who are excellent players, so obviously their methods have served them well. My opinion is that the real "support" is air speed. I've never taught anyone to consistently play with your belly pushed in either direction. To me, this seems rather unnatural and would make playing significantly more stressful than necessary.
My general feeling is that all the support you need will come from taking the breath that correctly suits the phrase you need to play. The more you fill your lungs on the inhalation, the faster the air will come out on it's own. I think of the lungs as being balloons. If you fill a balloon to 95% of its capacity and release the air, the air will come racing out at a high rate of speed without any help as a result of what I call "natural compression". If you fill a balloon to 30% of its capacity and release the air, the air will come out a much lower rate. The only way to get this balloon to release the air more quickly is to squeeze the balloon, adding artificial support to the release of the air. I view my lungs the same way. I aim to fill my lungs to a point that the air comes out naturally, without artificial support (pushing your belly inward or outward), at the speed that is desirable for the phrase.
Only in the extreme upper register, and usually the extreme upper register and dynamic register, do I feel that artificial support is needed. For example, when going for the high Eb in Symphonie Fantastique or the high D in Brahms' 2nd, the natural compression of the air is not enough to get the high note and the extreme dynamic. A little support from the abdominal region can help raise the air's rate of speed. There's not much of a better way to explain how this is done except to say that the same muscles you use to go to the bathroom are basically the same muscles you will use to reach this extreme range at a high dynamic.
As for always playing with your belly outward or inward... I honestly don't understand this. First of all, your lungs are not in the same area as your stomach. Air goes in your lungs, your lunch goes in your stomach. Pushing your lunch in or out, in my opinion, isn't going to make you play better. Just leave it wherever it naturally falls. Trying to correctly control your breathing apparatus while focusing on keeping your stomach in or out seems like a lot of wasted effort that will only have negative effects on your playing.
As always, this is the way that I play and the way that I was taught. I know there are many different schools of thought on this, and there many different ways of playing well. I hope this information helps!