I hope your English teachers' therapy sessions are going well.
Don't you mean "teacher's"? The other question - "for whom?" - is prompted by my recollecting that, a short while after a move on Zemlinsky's part meant that the precociously talented teenage Korngold's studies with him had to come to an end and Korngold began studies with an evident academic dullard whose name I cannot immediately call to mind so I'll refer to him here as "×", an exchange of correspondence between the two composers included a letter in which Zemlinsky asked his former student about his studies with × by saying "how is it going? is he making progress?"...
Anyway, to return to the topic(!), the problem here is, as has already been noted, that one can only really write about those who have a recorded legacy of improvisation; not only Bach but Mozart, Beethoven, Liszt, Franck and Reger, among others, are documented as highly accomplished improvisers but the only evidence that we have for any of it is in documentation (which is not, of course, to suggest that it wasn't true!). Traditionally, improvisation has perhaps been encouraged rather more in organists' studies than in those of pianists and other instrumentalists and the element of improvisation in jazz performance has often tended to have a largely separate existence from the world of "classically trained" performers (with certain notable exceptions, of course).
Among pianists alive today, perhaps Gabriela Montero is the most noted for including improvisations in recital programmes.
Best,
Alistair