Excerpts from the Complete Rule from NFL.Com published 9/24/23.
NFL emergency third-quarterback rule: Questions and answers
Q: What is an Emergency Third Quarterback?
A: A club may designate a bona fide quarterback as an Inactive/Emergency Third Quarterback for a regular-season or postseason game.
The player must be listed on the Gameday Administration Report that is submitted prior to or at the 90-minute meeting, subject to the following:
The Emergency Third Quarterback must be on the club's 53-player roster; the player cannot be an elevated Practice Squad player.
The club's starting quarterback (QB1) and its backup quarterback (QB2) must also be on the club's 53-player roster.
A club must have Two Bona Fide Quarterbacks on its 47/48-player gameday active list to designate an Emergency Third Quarterback.
A club cannot designate an Emergency Third Quarterback if it has three or more bona fide quarterbacks on its 47/48-player gameday active list.
*Q: When is an Emergency Third Quarterback permitted to enter a game? *
A: An Emergency Third Quarterback is permitted to enter a game only if the club's two active quarterbacks are unable to play either by disqualification or injury, including for example, if QB1 is unable to play and QB2 is being evaluated for an injury.
Q: Are clubs allowed to play an Emergency Third Quarterback in non-traditional fashion (i.e., Wildcat formation)?
A: No. An Emergency Third Quarterback who enters a game must line up under center or in shotgun and must take the snap.
No other player may take a snap if the Emergency Third Quarterback is in the game.
Link To Full Article Below:
https://www.nfl.com/news/nfl-emergency-third-quarterback-rule-questions-and-answers
A few things to point out as highlighted above in red. (Note: Swap Rattler for Haener in example below if you want.)
1. First and Foremost. As long as Hill is listed as a QB and/or considered a "Bona Fide" QB he counts as a QB on the gameday roster.
2. Carr, Hill, and Haener active on gameday would constitute 3 Bona Fide QBs and therefore prohibit the Saints from declaring Rattler as the EQB3.
3. The Saints could list Carr QB1, Hill QB2, and declare both Haener and Rattler inactive on gameday, with one of the two designated as the EQB3. This would mean that if Carr were to get injured, Hill would enter the game as the #2 QB and unless he were also to get injured, he would finish the game as the backup QB. The only way the EQB3 would enter the game is if both Carr and Hill were to be injured and unable to return to the field.
4. The Saints could change Hills designation as QB back to TE. They would then be able to start Carr, name one of the youngsters as QB2 on gameday and then make the other youngster inactive with a EQB3 designation. The problem with this is that if Carr were to get injured and the designated QB2 got injured prompting the EQB3 to enter the game, Hill would not be able to take any QB snaps or Wildcat type snaps for the remainder of the game. Highly unlikely, but still a point.