A textbook example of the latter approach to real emergency management is the 1989 airline emergency landing in Iowa. If you have seen the movie and other training videos based on this incident, there is no doubt that there was one person in charge. Coordinating?
Of course, but with clearly defined authority and responsibilities. Such a person speaks and acts on behalf of the chief executive of the jurisdiction.
Based on this brief and simplified description of emergency management, you can determine where your community stands. It is up to you to function as effectively as you can within the reality of your organizational structure and political facts of life.