Identify and explain the various types of database users and their roles in a database system. Choose one type of user $($e$.$g$.,$ end$-$users, database administrators, application programmers$)$ and elaborate on the specific tasks and responsibilities associated with that user role. Discuss how the interaction of different user types contributes to a database system's overall functionality and efficiency. Finally, provide a real$-$world example or case study where the collaboration of different types of database users played a crucial role in the success or failure of a database project. Ensure your response reflects a deep understanding of the roles and responsibilities of database users.