With all do respect @ryan I am struggling with this comment. I would define scrum as a lightweight project management framework that can be applied in any industry and we are starting to see that. The scrum master is a servant leader who helps their teams achieve their goals. Therefore, they have many different responsibilities such as motivator, facilitator, scrum champion, team protector, and impediment remover. Overall, they should be looking at the empirical data to drive continuous improvement. So I would say it goes beyond facilitation of code.
There is a section in Michael James Scrum Master checklist that is labeled "what am I doing to help my organization." This is because Scrum Masters are true change agents, looking beyond the team and not accepting organizational constraints. This is why a scrum master is an agile coach.
I once worked in a small company with experienced scrum masters. We did not have coaches because they were not needed. Scrum Masters did the coaching. I believe this is how Ken and Jeff envisioned the role.
Two years ago, I had an engagement at a small company in which was new to scrum with no experienced scrum masters. I coached four teams and new scrum masters. My main focus was the scrum masters because they were going to be the coaches when I left. This is the only way a transformation can be sustainable after coaches walk out the door. I stay in touch with them all, playing a mentoring role, encouraging them to get involved in their community to improve at their craft and accept the role of change agent.
In my current engagement at a mid sized firm my title is team coach. I sit with the teams and work with them daily. However, I meet with leadership regularly and are providing coaching on all levels. I have made it very clear that the more we scale, the more risk we will accrue of not being able to sustain the transformation. We are reaching the tipping point of me being stretched too thin, so more coaches are on the way.
My point is context is everything. In the world of Agile at Scale, Lisa Adkins and the Scrum Alliance are catering to a real world problem and trying to provide direction on how to tackle it. And making money doing it:) I do think dilineating the different coaching responsibilities at the different levels is a good idea when in large enterprises and too heavy in small shops.
My final point is all agile coaches should be able to coach on all levels. The common theme in the answers above Is that coaches focus on improving the agile mindset of people, teams, and organizations. They can play different roles within the organization.