Please see the next section below what you quoted:
"The second player has not predetermined if their character will believe or disbelieve. The player wishes to find out what their character experiences so that based on that they can make a call. They need the full description of the environment including the results of their passive Insight (PHB, pg 175) in order so that they can determine what their character thinks and does."
In other words, the player has built a character, expects to have a chance to tell if someone is lying, NPC or PC, because that's a common and legitimate passive usage of the Insight skill, and is asking for the DM to describe if they noticed something.