There are a good deal of other elements as well, though the owner to player hierarchy is roughly what you describe. Not every team has the exact same organization - for example, some teams the Field Manager might also make some player personnel decisions, or in many teams there is a General Manager but also a VP - Player Personnel who is above the General Manager (basically a promotion path for General Managers).
However, baseball teams have a major part you ignore here, which is the scouting and player development staff. Scouts work to identify players that the team might target, either college/high school players to draft, or minor league or major league players to trade for. Player development personnel, on the other hand, work with the minor league teams that are contracted with the major league team, to develop those players, identify those who are ready to move up, and identify those who can be traded for other personnel. This department is generally the largest in terms of employment (other than park operations, who would employ the retail and food service staff).
There is also some differentiation on the coaching side; there are the coaches who work on development, but there are also the game-day coaches. Only the pitching coach really has a role in both (as they work with both the pitchers during non-game periods, and also go on the field to adjust strategy and help calm pitchers); most other development coaches have limited game-day roles. However, there are four game-day coaches who primarily exist for that game-day role; the bench coach, the bullpen coach, the first base coach, and the third base coach. Those coaches have specific jobs on game day, and might not have any other role (though they may also have a development role separate from their game day role, or may not). First and Third base coaches direct players when they are runners on the field; the bench coach manages the players on the bench and ensures they're where they should be, gives them scouting information, as well as taking over for the manager if they are incapacitated or ejected. The bullpen coach manages the relievers in the bullpen, having them warm up when requested.
Finally, there are many others who make sure games are possible who don't have a direct effect on players. Field ops, park ops, ticket sales, etc.; there are quite a lot of people in these different roles that are important to the game occurring, but don't actually interact with or impact players directly.