“Why does a baseball team have a manager instead of a head coach?”

I answered a similar question with this: Max Steiner's answer to How did the nomenclature for the leader of a Major League baseball team become "manager" when the designation for every other sport -- including nonprofessional baseball -- is "coach?"

The difference in the terms stems from the origin of the sports themselves, and who began playing them. Baseball traces its roots back to “sporting clubs”, groups of gentlemen who would get together for competitive matches (as an alternative to, say, forming into battalions and fighting in mortal, martial combat). These clubs did not require any specialized skills, nor any specialized training to progress and diversify those skills. Without a need for training, there wasn’t a need for coaching.

What the clubs did need is someone – one of their own – to coordinate matches with other clubs, to determine the roster and lineup for the coming match, and to interact with umpires (or referees). Certainly at the start, and later, more often than not, this manager was a player, at least active and eligible to participate. Gradually, those players would get older, or less active, but were valued for their loyalty, dedication to the club, or effectiveness in managing the club’s game-to-game operations. Without roster limitations, these managers could still be eligible to play, but most would have an arrangement with the club’s ownership to not actively participate, but instead manage and mentor the active players.

It’s from this mentorship where coaches were introduced. These were experts, former players themselves, who would be imparting their experience upon players on the club for various aspects – pitching, hitting, fielding. Coordination of these efforts and activities would flow through the Manager, who would in turn be responsible for arranging the lineup, making pitching decisions and changes, and ultimately answering to the club’s ownership as to how the club was functioning.