ppianista, the differences are actually coming about because each time I write something in this thread, you have argued against what you thought I was saying. Your views are probably the same as mine, but when you read what I say, you have argued against it... hence, what you *thought* I was saying. I have a pretty good idea of where this misunderstanding is coming from so please read on.
I have in mind your post in the thread that I started. In it we're all discussing the teaching of technique, which is a component of music. There you talk about the beginner who has all kinds of "wrong" things in his mechanism, but you allow them, and gradually shape what the beginner can do. Reading this, I see a teacher who watches his students and tries to shape their abilities over time. Therefore you are already there. Inherent in this is the fact of skills plus knowledge about music, combined with playing music - because how other than playing music do you get a context for those abilities? And without abilities, how do you start playing more refined as you progress? They are two co-dependent things. So you're there, where my thinking is.
There is something missing when you read my writing and disagree with it, and I think I know what it is. You would be familiar with how you teach, and often good teachers will think that how they teach and maybe how they were taught is how everyone teaches. I've encountered this before. When I write, I am mindful of what adult students frequently encounter over here. I'm writing "over here" because with the Haenschen Klein reference, I suspect you're "over there" on the other side of the ocean.

Actually what happens "over here" is not only reserved for older students, but maybe more frequently for older students.
So about the "over here":
I emphasize skills and knowledge because often they are almost totally absent in the equation. The common idea is that adults do not really want to learn to play the piano. Rather, they want to manage to produce their favourite music without spending much time practice, or taking lessons for very long. While you might give music, and through the music try to gradually shape technique so that the music can be honed ---- and while you might give "theoretical" knowledge like learning to read and interpret music through more understanding --- and this happens hand in hand WITH the music ----- this requiring a teacher who is aware of these things while teaching ---- That part doesn't come in. Ever. Because it's deemed adult students don't want to really learn how to play. A decade or so ago you would even read, "Give them whatever they want, because they soon quit anyway." There was some fire with that smoke.
About "children's methods" and "adult methods". The method book is probably a North American phenomenon. This is not about whether you play a simplified version of Ode to Joy versus Alle Meine Entchen. It has to do with what is taught, and what goals are set. By "what is taught" I am not talking about repertoire, but about the skills and knowledge part. Contending with key signatures and modulations is a "what is taught". Getting a feeling for chromaticism, or learning how to use the pedal, or becoming familiar with music of various periods is "what is taught". And along the way, there are also pieces.
I you have not examined the method books, and the "adult" counterpart by any publisher, you won't know what I'm talking about. And if you don't know what I'm talking about, you risk arguing against what I'm saying without knowing what I'm saying. That is what probably happened.
The same with the new phenomenon of capturing the "adult market" by creating programs in conservatories for rank beginners. Of course a teacher can give any kind of music, and can teach interpretation. But if it's a rank beginner, then the teacher has to be aware that the beginner cannot just up and emphasize the main voice in polyphony. The student first has to learn to read notes, learn how to practice, learn how to control the most basic voicing. In other words, the teacher has to develop the basic skills of the student. And a lot of the teachers used to teaching advanced students don't know how to do that... don't even think of it.
These are things I'm seeing.
And if THIS teacher is actually improving her student's skills, then the right thing is happening on that front. I do agree that the comment about "never being good' is not a good comment. The trouble is that finding any teacher at all who will give skills along the way and knows how to do so, AND is willing to take on adult students is very, very, very difficult. That is the reality here.
If you began lessons as a child you won't run into these things.
I hope this is a bit more clear.