It is important when discussing concepts to make sure both people are picturing the same thing. What I perceive as "practice" may be different than what you perceive, CJP, and I don't know what happens in the lessons you teach, just like you don't know what happens in my lessons. That should be straightened out before talking past each other.
Some examples:
In a recent lesson we worked on a technical issue. I was feeling uncomfortable in the way I was trying to play certain chords in the lower register of the keyboard, and a there was a figure in the third beat of each measure that was a strain to play. My teacher worked with me, saw what I was doing, suggested some other things, and guided me in how I was leaning on the bench, what I was doing with my hands, wrists, fingers. When I got where I needed to be, he had me do this for long enough under his observation that we were both sure that I could do this at home. We did some other things, but this was the main part. It depends on what I'm learning.
In my practice at home, one of the things that I did was to work on these technicalities in the way I had been shown. I did them separately, and then I applied them to that passage. I break my practising into mini-sessions, each time focusing on something else. It might be particular passages of the music, but it will also be an aspect: smooth fingering, or dynamics, or whatever. Maybe 20 minutes on one thing, 40 minutes on another. I'll be studying the music. I'll be working with chords and inversions. I may invent a mini-exercise to help with a difficult spot.
The practicing does not resemble what I described in the lesson. When you practice you have to do a thing long enough, and you vary the angles. In a lesson you couldn't get at much if you went at it for as long as you do in practicing. You are working with a teacher and learning how to do new things, or how to do things in a better way. Sometimes you are problem-solving. In practicing you then spend time doing these things. It is not the same activity.
About performance and practicing toward performance. I am learning how to play the instrument and I am learning about music. So in my learning phase, I may study chords, analyze the music, learn about the composer and the genre. I may spend time learning how to do a good staccato. I may take sections of music that are difficult and work only on them. Obviously I won't perform that way. I will not perform measures 25-27 played backward, followed by measures 2 - 7, with a break in between, with some time in between playing arpeggios in D major. If I am practicing toward performance then I WILL play my piece from start to finish. At this point I have whatever abilities that I have gained, and now I'm consolidating the piece.
I don't know if anyone else does things this way, but this is the kind of thing that I mean. I don't think that I am alone, though I also think that there are at least a dozen variations of what people do.
** When you say the practicing is done in lessons and then at home, can you give an example of what kind of practicing you mean?
Topic: Link between sight reading and performance