Thanks for a good article, Alexandre!
Based on my experience, I can say that some companies prefer to hire “Agile Coaches” instead of “Scrum Masters” just because they are not sure that Scrum fits them well. For example, there might be 2 teams who share one Agile Coach, where first team follows Scrum and the second one — Kanban. So, calling this person a “Scrum Master” would sound weird.
At the same time, it’s almost impossible to create a successful agile team in an old-fashioned company and vice versa, so at some point every Scrum Master has to deal with issues on the company-wide level, not only at the team level.
Right now I describe the difference to myself that way: Agile Coach is a position in an organisation and Scrum Master is a role in a Scrum Team.