I like your advice, about being a client, I think I can apply it with my first teacher.
About the second, it is only That I am becoming a profesional musician and she is my teacher formally, where I need to get good grades or I will get expelled, can this advice be applied there as well?
Being a professional musician is not like being, for example, a physician or a police, where you cannot work unless you have passed a certain education with approved results. Musicians belong to the artistical genre and there are no licenses for artists. In the end, it does not matter what their "papers" look like, which exams they have or not have passed.
If you are a dancer you go to auditions and if you dance well, you will get the job. They don't dismiss you when they see that you have not been to so-and-so dance school. If you write an excellent novel, the readers couldn't care less about your formal education as a writer.
I do not mean that formal education is pointless; it could of course be of the greatest help. But it is NOT mandatory, and therefore your whole future as a professional musician does not rely on you being expelled from a school because you and the teacher did not get along.
Besides it is not good for a school's reputation to kick students out because they are getting nervous in lesson time and do not improve in this aspect. That says far more about the school than about the student, and other students will hesitate to go there, which means the school will be out of business rather soon. They need students who graduate with great results even if they were bad at first, and who happily recommend the school to others.
So I think you should be very honest to your new teacher and spend some time talking about your concerns and how you feel. If you cannot trust her discretion and her will to help you, you will do better without her. But give her a chance first. And if you are too shy to have this one-to-one-talk ... ok, now I must be a bit harsh. I am not a professional musician myself but I know quite a lot about that world, and the world in general. If you are going to be professional, you simply have to be bold. You need elbows and thick skin, you must be able to face people who are not nice to you, who give you harsh and sometimes very unfair critizism, who are difficult to work with, and you must be able to communicate in a clear, honest way. I don't know how old you are, and of course you are not required to be more experienced and tough than your own grandpa before you are not even an adult, but be realistic. You cannot use "shyness" as an excuse in long terms to avoid potentially unpleasant confrontations, not if you are going to choose this career.
Besides I don't think the conversation will be unpleasant at all. Your honesty will be appreciated, provided that the teacher is sane.
PS. And to add some perspective to it: I gave up the idea of becoming a professional pianist even way back in my teen years, because I had a major problem with stage fright. (I still have.) I simply found recitals terrifying and then I had to ask myself why I should choose this torture as a profession.