Final answer:
The common element in all four definitions of Sports Public Relations is media engagement, which involves communicating with the media to shape the public image positively.
Step-by-step explanation:
When looking at the common elements in all four definitions of Sports Public Relations, we focus on activities that are central to the role regardless of the specific field of application. While activities such as social media management, fan engagement, and player contracts are often associated with sports public relations, it's media engagement that is fundamentally a part of all four definitions of sports public relations. This encompasses efforts to communicate with and through various media outlets to shape and promote the public image of sports entities, whether teams, individual athletes, or sporting events.
Unlike journalism, which is dedicated to reporting news to inform the public without bias, sports public relations focuses on positive promotion. Press releases and paid advertisements are common public relations materials used to generate favorable publicity. While social media management and fan engagement are also significant components, they are not mentioned across all definitions as consistently as media engagement.