You are right to identify that Bierce uses narration in a very interesting way in this famous story. For example, in section 3, the point of view shifts to the third-person-limited point of view seconds before the actual death of Peyton Farquhar. This of course is appropriate given the way that this section explores Farquhar's desperate flight of imagination. Such a detached perspective allows the narrator to maintain a realistic stance, even as Farquhar's mind is obviously running away from reality.
You might like to think how the story starts by being written in the omniscient point of view. In addition, the beginning of the story is well known for the way that it presents the opening scene almost as a film might present it:
A man stood upon a railroad bridge in northern Alabama, looking down into the swift water twenty feet below. The man's hands were behind his back, the wrists bound with a cord. A rope closely encircled his neck. It was attached toa stout cross-timber above his ead, and the slack fell to the level of his knees. Some loose boards laid upon the sleepers supporting the metals of the railway supplied a footing for him and his executioners--two private soldiers of the Federal army, directed by a sergeant who in civil life may have been a deputy sheriff.
Note the very visual nature of this description and how the scene is set, introducing the main character and providing us with lots of detail. Of course, as the story progresses, as previously noted, the point of view changes as we zoom in on Peyton Farquhar, and his feelings and emotions, combining flashbacks that describe how he managed to arrive in this situation with his own delusory flight of fancy before he dies.