The Agile Coaching Institute has a wonderful breakdown of the skills that are needed to successfully coach agile teams. The model is useless for the work that I do, however, and I am an agile coach - a transformation coach.
Consider basketball. When pro basketball players think about their sport, they think about the things that they experience, and that are important to them: the plays during the games, the practices, the endorsements. Aspects of basketball - aspects that are central to the sport, such as sponsorships and team management - are on the periphery of a player's thinking.
But if you were to ask a team owner what things matter, they would have a very different perspective. The two perspectives are compared in the figure below.
The same holds true for agile coaching. Agile team coaches experience activities pertaining to their teams, and aspects of software development that are external to teams are experienced - by team coaches - in terms of the way that those things interface to the team; those externalities are experienced as somewhat peripheral. It is kind of like the famous "View From NYC" picture from New Yorker magazine that we have all seen: to a New Yorker, the features of New York loom large, but the features of the rest of the world - while just as important - diminish toward the horizon. In other words, one's perspective depends on one's experiences.
That is why "transformation" is only a single item in the Agile Coaching Institute's list of skills needed for agile coaching: because to a team coach, transformation is just one thing going on, and it is not usually central to what teams think about. Teams are affected by a transformation program, and they participate in it if it exists, but it is not what they focus on each day.
In contrast, an agile transformation coach thinks about transformation every day, and their view of coaching - transformation coaching - is very different from the view of a team coach. The two views are illustrated in the figure below.
The two diagrams are linked: in the team coach's view (adapted from the Agile Coaching Institute's list of core coaching skills), "Transformation Mastery" is a single slice of the pie. In the transformation coach's view, "Transformation" is the whole pie; and in that pie, "Team Coaching" is a single slice. Thus, one set of skills does not subsume the other; rather, they inter-connect.