tl;dr The titles used may or may not reflect any specific skills or differentiation of skills in the services provided to an organization. High quality Scrum Masters are more well rounded than just knowing the Scrum framework.
One of the most common failures of agile product development efforts is a lack of understanding of the philosophy. Misunderstandings result in impediments and loss of value: organizations do not properly support self-organizing teams; customers do not collaboratively engage in the process; long term, plan driven mentalities remain over using emergent, value driven approaches.
Agile Coach should not be only about Scrum
Because the Scrum framework requires the role of Scrum Master, it can be a specific position to be filled. Others require or recommend different or additional positions. The position of Agile Coach is not, AFIAK, required or defined in any. The use of an Agile Coach to implement Scrum can be a differentiator of an experienced Scrum Master to help train and coordinate multiple teams each with their own Scrum Masters; sometimes it is due to Scrum In Name Only.
However that does not also mean that a Scrum Master should only be about the Scrum framework. Having knowledge of and experience in multiple systems, frameworks, methods, and processes results in a more valuable and effective Scrum Master.
Scrum ... functions well as a container for other techniques, methodologies, and practices.
Scrum Masters cannot and should not be solely focused internally at their Scrum Teams. Looking at the Scrum Master section of The Scrum Guide, three of the five opening sentences are about everyone and not just the Scrum Team; the first sentence being the one that you quoted.
The Scrum Master is responsible for promoting and supporting Scrum as defined in the Scrum Guide. Scrum Masters do this by helping everyone understand Scrum theory, practices, rules, and values.
The Scrum Master is a servant-leader for the Scrum Team. The Scrum Master helps those outside the Scrum Team understand which of their interactions with the Scrum Team are helpful and which aren’t. The Scrum Master helps everyone change these interactions to maximize the value created by the Scrum Team.
The rest of the section is divided into three subsections of servicing three separate areas of concerns: the two additional Scrum Team roles plus one for the organization at large. Based on the five bullets for supporting the organization, there should be little to no difference in how either role would provide valuable service to the organization.