The ScrumAlliance defines a Certified Scrum Coach as the expert in Scrum and has guided multiple organizations through the transformation. Recently, the ScrumAlliance featured a post by Geoff Watts discussing "What Really is an Agile Coach?", where he concludes in addition to being an expert in Scrum, effective coaches must also be certified by the International Coaching Federation, (ICF). ICF further distinguishes coaches from consultants:

While consulting approaches vary widely, the assumption is the consultant will diagnose problems and prescribe and, sometimes, implement solutions. With coaching, the assumption is that individuals or teams are capable of generating their own solutions, with the coach supplying supportive, discovery-based approaches and frameworks.

It makes sense to me, that organizations should hire a coach to help with the agile transformation. The Agile Coach should work with the highest leadership levels in order to engage them and understand what those senior leaders want for the organization and its progress along an Agile Journey. However, an Agile Coach cannot work alone. Team Coaches are needed to assist with that transformation.

Here's why and here is the sports analogy:
When learning a sport, in our youth, usually, or even when older, learning a sport requires direct guidance and instruction. This coaching is prescriptive, focuses on fundamentals, and provides immediate feedback on what the observed behavior to either reinforce or discourage. As we progress in learning a sport, we advance, and our coach works to refine techniques. The mentorship focuses on more advanced skills and more complex behaviors and habits. For the few that reach a level of expert acuity, maybe those who are professional athletes; the coach helps the athlete reach performance goals the athlete is setting for themselves.

In short: the Little League coach is prescriptive while the coach for the National or Olympic Team is there to help the athlete figure out their own solution and provides a supportive framework of discovery.

If your organization is going through an agile transformation, you want a variety of coaches to help along the agile journey. Team coaches assist the scrum masters, product owners, team members, and others. Team coaches should be prescriptive for teams who are starting out, new to agile techniques, and just beginning to understand the fundamentals. Mentorship is required for those teams who are further along a maturity or change scale. Team coaches know how to assess a team and recognize when to be prescriptive and when the team is calling for specific help.

A different type of coaching is necessary for management and leadership. Agile Leadership Behaviors are more challenging, because they are different roles and levels within the organization than those who function on a team. An Agile Coach recognizes that leadership also needs assistance in the transformation. Since executives want their organization to adopt the agile framework, there is some learning and change included in how they view and interact with that transformed organization. The Agile Coach is there to provide a supportive framework to assist the organization reach the goal it is setting for itself.

When the entire coaching team works together as an Agile Transformation Team, the results are stronger because of identification of the feedback loops and challenges that exist across the whole organization.