I'm an adult student and piano is my 2nd instrument that I'm returning to decades after playing as a self-taught teen. My insights come from two places: mutual support with another adult student halfways across the world with candid discussions of every possible issue and obstacle coming our way, family and friends who are professional musicians or en route. Among the insights:
We do not perform for our teachers. We are working mutually together with our teachers on a project: building our skills and building the piece or study we are working on. Therefore we can look dispassionately on this outside object the way a carpenter's apprentice may look at a chair, without emotional involvement.
In academic pursuits we must present a perfect paper. In musical pursuits it will not be perfect because it is not a finished product: it is a thing in the process of being built. Mistakes are wonderful: that's what we get to improve so that we become better and better.
The mere knowledge of these two points, and coming into a lesson with these two attitudes makes it possible to be at ease in a lesson, and the difference can be like night and day. Imagine the opposite of what I have written, and the effect of that opposite.
It helps to know that a teacher can see beneath our performance, what our practicing might have been like. Some residual effect may remain. The thing is that if someone practices 3 hours per day, they have 18 hours behind them by the time they come into the studio, and maybe 15 minutes to demonstrate what they have gotten right during the week - If they blow it, then it has all been for naught (especially if the first two points aren't there). The very fact that this is so creates pressure and nervousness. The most likely time this will happen, and the most frustrating, is when we have reached a plateau and everything is perfect and we're excited about showing our teachers what we have achieved. That is going to guarantee failure, and the resulting disappointment can destroy the rest of the lesson. It helps to care a lot less, and about something different. It also helps to know that our teacher can see, even in a flawed performance, signs of what and how we have practiced.
Knowledge of teacher expectations: You guys do not want us to perform a perfect and impressive piece. If you wanted that, you'd attend a concert and listen to a professional performer. You want us to master the skills you are trying to impart to us. The realization and resulting frustration in hindsight is almost comical, "You mean, that's all he/she wanted?" Your brilliant, "expressive" playing does not as much as raise an eyebrow, but you've played in time, able to count from one to four, and your teacher is over the hills with happiness. Grrrr. Oh, ok, lightbulb time. The thing is that if you do think you are to do impressive performances, and you do miss the "counting to four" moment, that's stressful as the student strives and strives.
On the same subject: not being able to count to four is not embarrassing, it is not a failure, it is not something that a four year old can do and therefore humiliating for an adult not to be able to do. Believing that it is causes unnecessary stress.
The pedestal on which we place you, the Teacher, The Musician. Clay feet inevitably follow. After that, there is a human being who has worked incredibly hard to achieve what he can do, and if I, as a student, wish to achieve, I have to work as hard to reach my own potential. The final stage is probably true respect and appreciation based on a bit of knowledge. It comes in stages if you can get that far as student.
A nasty little thing comes up if you're doing a good job teaching and we're doing a good job following you: We start hearing how we really sound instead of how we imagine we sound. The results can be devastating because to our own ears we have suddenly gotten worse and can't improve no matter how hard we try. If you tell us that we're improving, then you must just be sweetening the blow and protecting us from the awful truth. Kids don't become self-aware until around age 12 or 13 so it doesn't hit them the same way. And possibly they don't care in the same way that an adult dues.
Something a bit more abstract is the concept of music making itself. Music performances are perceived as a whole, but are not created that way. They are built element by element, apart and together, and they improve over time in practicing and rehearsing. If you expect that it is either good or not good from the start, and want it all to be perfect immediately, that is stressful. I have run into several adults, including my own parent, who believed themselves to be failures because it was not perfect immediately. Creating a piece of music is more like creating a sculpture. You begin with a blob, and whittle away, turning it round and round, until eventually you have some cylinders, spheres, cubes. You whittle some more until eventually it is a recognizable figure - a child holding an umbrella. No sculptor would stare at the blob, or the spheres and cubes, and despair, because he knows that sculpting is a process. We don't necessarily know that, and this is another reason for stress.
If I know that I am creating a sculpture, and also know that I am the sculpture being created in the sense of being given skills and knowledge which cannot come all at once, my perceptions will be different. The bottom line is perception and attitude which cause the stress and uptightness. Ultimately this lies in the hands (minds) of students, and your reassuring words and positive manner are only part of the equation. If you tell me I'm doing well, I have to be able to believe you and know what that means.
In my personal journey I have found my dialogue with a few wise students and ex-students to have been very helpful, as well as some candid talks with a teacher or two. hth