I've also worked in consulting in a past career, and I've had a hired coach, and the line you quote with words like "partnering" I would describe as part of marketing the product. They're not exactly going to say "hire us to push you out of your comfort zone", but that is the role.

You said: "they keep asking questions and pushing" -- that's what I mean by setting the agenda. As I mentor, I don't see my role as "pushing". Questioning, sure. Providing perspective, sharing my stores, yes. Actionable feedback on skills is the closest I'd come to "pushing" and even then, they can take it or leave it.

When I see people talking about coaching, I often see -- directly or indirectly -- some aspect of the role of the coach to be to "bring out their best". I rarely see words like that used to describe mentoring relationships.