That is in direct conflict with the game's definition of what a passive check is
Respectfully, I don't think so; if I had said that a die is rolled for a passive check, that would be a direct conflict with the book definition. What I would love to do is simply to find a convincing narrative case for a Passive Investigation check, which is the whole point of this thread.
I think that it's fair to say that the narrative case for rolling a die is the intention of the character, as stated, thusly:
The DM calls for an ability check when a character or monster attempts an action
...'attempts,' being synonymous with 'acts toward an intended result.' Then, by deduction, events that occur without player intention would be what initiate a passive check. Restated, 'Active' indicates that the character intends to do something, and 'Passive' indicates that they don't. To be fair, I do think that there's a problem with my phrasing; rather than suggesting that a character could juggle without trying to juggle, or stealth without trying to stealth (even though I think they could), it would be better to describe this as 'events occurring around the character without the character's intention to influence said events.'
The real-world model for Passive Perception is exactly that, right? That a player, intending to observe and notice, would roll a die, in an active check, but whether they notice something despite a lack of intention would be that passive check.
If there's no such thing as stuff happening around the character, without the character intending to influence them, as the narrative context for roleplaying, when does this game mechanic ever apply?
The PH does provide one sound alternative, in a passive check basically being an average of what would ordinarily be a long series of the same active check, such as making a Passive Investigation check, instead of lots of Active Investigation checks for a day of library research.
Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster.
...BUT, it also has that 2nd example, of noticing a hidden monster, which still leaves the question of what the difference between Active and Passive is, for the character's story. To notice a hidden monster, why wouldn't the DM just ask for an Active Perception check? My answer is, that the character just happened to notice, without being on high-alert, without knowing that something is out there. ...without intention.
All this is only a suggestion. If the ghost of Gygax chimed in, and called me wrong, I would totally accept that, then give you a blue ribbon and a cookie. Absent his response, I would like to make this suggestion.