Good write-up, Tim. To build on that, I like how the Agile Coaching Institute defined these four stances:

Coaching Ability to act as a coach, with the client’s interest determining the direction, rather than the coach’s expertise or opinion.

Facilitating Neutral process holder that guides the individual’s, team’s, or organization’s process of discovery, holding to their purpose and definition of success.

Mentoring Ability to impart one’s experience, knowledge and guidance to help grow another in the same or similar knowledge domains.

Teaching Ability to offer the right knowledge, at the right time, taught in the right way, so that individuals, teams and organizations metabolize the knowledge for their best benefit.