A perceptive question! The phrase that you refer to occurs twice in Chapter 12:
Everyone behaved with complete presence of mind.
Several paragraphs later, we have
Everyone behaved with complete presence of mind and that included Phineas.
Phineas has just suffered a devastating emotional blow when finding out the truth of the "accident," regarding his fall from the tree. From Leper's testimony, it becomes obvious to all that Gene had to have been the one to have caused it. Gene knew that he was responsible from the start, but Phineas had refused to accept it because Phineas only sees what is good in others. He reacts by attempting to run away, only to fall down the stairs and injure himself again.
When faced with Phineas' physical injury, the boys act with "complete presence of mind." They can quickly spring into action, getting the doctor, notifying Phil Lantham, keeping Finny still. Everyone is quite calm, even Finny. In contrast to the chaos caused by Leper's testimony, all is order and efficiency here.
It's as if all the boys have moved on in some way, finally accepting the reality of the world in which they live: the reality of betrayal, war, nervous breakdowns, permanent injuries, and their own dark sides, which will prepare them to accept the reality of death. Phineas' second injury, unlike the first, is one that they are ready and able to deal with. We see the boys and the school officials at their best. This efficiency paves the way for the final reconciliation between Finny and Gene. Finny forgives Gene and accepts the truth: "it was just some ignorance . . .some crazy thing . . .something blind, that's all it was."
The novel is a journey from the innocence of youth to the darkness of human experience. The boys develop the "presence of mind" to face and accept this transition.
As the narrator, Gene thinks about others' presence of mind for several reasons. First, when Finny suffers his second devastating fall (in Chapter 12, the fall is down a staircase), it is obvious to everyone that he is injured badly. The boys are able to go from being divided over the Gene's "trial" to being able to rally together to get Finny help. Similarly, Dr. Stanpole and the infirmary staff get Finny comfortable fairly quickly and even ask Gene to get Finny's stuff from the room.
These literal examples of presence of mind are important, but what that phrase means to Gene personally is more significant. Throughout his time at Devon, and especially after Finny's first fall, Gene has searched for peace of mind. He tries to figure out why he did what he did to Finny. He worries about whether the other boys know what actually happened at the tree. Because he is looking for his own separate peace, he marvels at others' ability to think on their feet and to do the right thing quickly. To Gene that is the presence of mind that he wants, a mixture confidence and charisma.