When I was a student at Manhattan School of Music, I had a teacher who had been a fairly famous pianist at one time. A student was considered, by most people I encountered, to be very lucky to have the chance to study with him. He was a kind and wonderful Austrian gentleman and certainly a very accomplished pianist, and for a time, at least, I adored him. But he didn't know how to teach (something I only realized towards the end of my four years with him, unfortunately). I'm not sure he really enjoyed teaching but he did it out of necessity as his own career in concertizing had waned. Possibly he enjoyed his few most talented and advanced students (I was not one of them!) who already played so well, that all he felt he had to do was help them with "finishing touches." (The problem I noticed, however, with his students as well as students of other teachers, was that the students tended to start playing too similarly to their teachers, instead of finding their own styles or interpretations. We could often identify who the student studied with just from hearing his or her playing.)
What I needed, however, was a real teacher, someone who could help me with technical challenges. When I played I had fatigue and pain, and I did not believe that was just something to get used to or "muscle through." If I stopped practicing a piece for a week I could no longer play it. These and many other issues continued to plague me after years of lessons.
Many pianists who are quite accomplished do not necessarily know exactly how they do what they do, especially in the realm of technique, and therefore cannot help others. (Like athletes, they may have an innate physical talent that enable them to accomplish certain feats somewhat effortlessly.) In fact, much of the instruction I was given on technique I now know to be quite wrong, or at least wrong for me. There is not just one way of doing things; a lot may depend on the size of your hands, your particular body type and musculature, and other factors. (Even things like fast-twitch and slow-twitch muscles, a factor in many sports, may play a role.)
During the lessons he sat several feet behind me so he could not even see what my hands and arms were doing. How could he possibly help me with technique if he didn't even see what I was doing? He had a one-size-fits-all answer to technique: hours and hours of particular technical exercises. Those exercises caused me pain and fatigue and did not increase my technique in the long run. Yet he had no other solutions for me.
If I had been at the point where my technique was more advanced, perhaps I could have benefited from his vast knowledge of the piano literature. He could have possibly helped me with "interpretation" of my pieces. This is what I would call "coaching," essentially fine-tuning an otherwise high level of playing.
Thankfully, after I graduated from the conservatory, I found what I had been seeking: a real teacher. His name was Joseph Prostakoff. He didn't concertize (he said he couldn't deal with the nerves) but his first love was teaching. He had mostly advanced students and mostly students like me -- pianists who knew that what they had learned was not necessarily right for them and who were seeking something completely different. (Several well-known pianists who had significant physical problems and even injuries had gone to Prostakoff to learn new ways of playing.) It's not an exaggeration to say that I learned how to play the piano all over again, from the very basics of how to use my body, arms, hands, and fingers, right down to the physics of how the sounds are produced. He sat next to his students at the piano and his eyes and ears were on "high alert" to pick up the slightest unnecessary tension or strain in our bodies, the rhythm off by a fraction of a second (due to improper technique), the minutest variation in dynamics which was not intentional (again due to improper technique). And most important, he had a vast array of ways to deal with all of these problems. He always said he did not teach interpretation; instead, he wanted to give the me the tools to say what I had to say through the piano. I remember after we had worked on a particular passage, and I had played it with a freedom of technique and phrasing and a beauty and power I had not had previously, he winked and said: "You see, it's not because you understand Beethoven any better, but because you changed what you did physically." He was not against thinking about or exploring different interpretations, but what he made me realize is that you can't just slap "interpretation" (especially if it is not authentically your own) onto a faulty basic structure, like a coat of paint to hide the flaws underneath.
This is what I would consider to be real teaching. It would be finding the way to play which physically enables you to achieve the sounds you want. (If you've read my posts on technique, exercises, and so on, you'll see this is why I advocate never playing mechanically; it trains you to use technique and movements that don't produce the sounds you ultimately want.) It does not matter how brilliant your "concept" of a piece is, if you don't have the tools to physically produce those sounds.
I often see "Master Classes" offered with some pianist/teacher, and students flock to them (often at very high prices!). Most of these would be what I would call coaching. You could go to one and the teacher might tell you to play a passage louder, while another might tell you to play it softer. One might say you need more rubato and another may say need less. You are receiving their interpretation, which may not sound or feel at all authentic for you. Interpretation is just that: highly individual. There are certainly some pianists and teachers who are so brilliant and so perceptive it would be worth hearing their insights. But don't confuse coaching with teaching. If you are studying with a teacher who tells you to "learn the notes at home" so that the lesson time can be used for "polishing" or "interpretation," you have someone who wants to coach, not teach. If you are studying piano with someone but you feel your playing is not really growing, that you have problems which are not really being solved or even addressed, or you don't quite feel that you are 100% yourself at the piano, you may have a coach when what you need is a teacher.