In a situation where B1 safely reaches first but leaves toward the dugout thinking he is out, what is the ruling?

In this scenario, the correct ruling is that the player is ruled out. If a player reaches first base and then leaves the base toward the dugout, thinking he is out, he is considered to have abandoned his effort to advance. In baseball, a player must be aware of the play and stay engaged until they are certain they have completed their objective. Since the player has left the field of play and is no longer in contention to proceed to the next base, he becomes an easy target for the defensive team to make a play on him. When the defensive team tags him or appeals that he has left the base, he can be declared out for abandoning the play.

It's important to note that simply reaching first base does not guarantee that the player is safe if they make a subsequent mistake, such as leaving the field of play based on a misunderstanding. The rules dictate that players must remain in their base area until play is definitively ruled on. Hence, the ruling of out is appropriately applied in this situation.