[SanDiegoSteve]

Jim Evans offers this:

"Customs and Usage: Until the mid 1970's, it was the umpires' responsibility to monitor the field for 1½ hours prior to the start of a major league game and submit a written report of any fraternization violation. Since that time, the task of upholding the integrity of the game in the public eye has become the responsibility of each club."

This is all the MLBUM has to say on the subject of conduct during the game:

"2.2 CONDUCT OF PLAYERS DURING PROGRESS OF GAME
No one except the manager, coaches (maximum of six), players and substitutes in uniform (including players on the disabled list), bat boys in uniform, two trainers, one member of conditioning staff, and one equipment or property person may occupy the bench or bullpen during the progress of the game. All other personnel must exit the bench or bullpen area before
game time.

Visitors are not allowed on the players' bench or on the field at any time during the progress of the game.

Players in the bench area may not sit in photographer stands adjacent to dugout during the game. During the progress of a game, all players and personnel of a club must be on the bench except when they have some duty to perform as coaches, base runners, batter, or on-deck batter. Players of the side at bat will not be permitted to fraternize with players in the bullpen either of their own or of the visiting club, but must come to their own bench even if, in their judgment, they are not likely to bat in the inning.

Only pitchers, catchers, players serving as catchers, and coaches should be permitted to stay in the bullpen during the game."