Q: What is a coach?

This is a question that I thought I had a good answer to. Now I'm not so sure. (It doesn't help that much of my original answer boils down to "coaching is the name I give to the thing(s) that I do".) I first felt unsettled as I read what Bill Caputo had written in an email: "Coaching is -- XP or otherwise -- about helping the entire team, define, understand, and improve the practices and processes of that team. As such it is a leadership position -- and also on the ground. I have said more than once, that the best coaches coach by achieving excellence in the practices they are teaching, and living those practices while working on the team -- including and especially the practice of pairing, which is the most effective way they can transfer their knowledge. More evidence that leadership itself is an activity and not synonymous with one's job description".
Could it be that what I call coaching is what other people call leadership? I'm not sure: leadership and coaching are definitely related, but while an effective leader can certainly be inspiritational, the "coach-as-inspirational-example" model seems to be missing something important. But I couldn't pin down what it was at the time.
Anyway a month or so after that I met up with some of my colleagues to discuss whether coaching really is 'just' an activity, and that led to more of a debate about exactly what it is a coach does, or can do (distilled here by Alan Francis). After the dust settled it seemed to me that some people were only interested in discussing coaching in terms of an activity that is done to other people. (The "coach-as-teacher" model.)
Whereas I see coaching very much as an activity that is done with other people, in order to bring about transformations in modes of thought and behaviour. This has become clearer to me as I have been reading up about a guest speaker who is coming to ThoughtWorks to talk about her (executive) coaching work.
This page for instance says: "the need for coaching comes from the absence of true conversation in the workplace" and "genuine human presence is the most subversive and creative thing. As a coach, be present in a way that speaks to the secret subtext of [the client's] potential". This definitely sounds more like the level on which I want to engage...
I'm sure that I'll be posting more on this topic in the future.