Here is what the FIDE Laws of Chess taking effect from 1 January 2018 have to say. There are two relevant sections, one for the general rules and one for special rules for rapid and blitz time controls.
Article 8: The recording of the moves
Amendments regarding electronic scoresheets effective from August 1, 2022.
8.1.1 In the course of play each player is required to record his own moves and those of his opponent in the correct manner, move after move, as clearly and legibly as possible, in one of the following ways:
188.8.131.52 by writing in the algebraic notation (Appendix C), on the paper ‘scoresheet’prescribed for the competition
184.108.40.206 by entering moves on the FIDE certified ‘electronic scoresheet’ prescribed for the competition.
8.4 If a player has less than five minutes left on his clock at some stage in a period and does not have additional time of 30 seconds or more added with each move, then for the remainder of the period he is not obliged to meet the requirements of Article 8.1.1.
Appendix A. Rapid chess
A.1 A ‘Rapid chess’ game is one where either all the moves must be completed in a fixed time of more than 10 minutes but less than 60 minutes for each player; or the time allotted plus 60 times any increment is of more than 10 minutes but less than 60 minutes for each player.
A.2 Players do not need to record the moves, but do not lose their rights to claims normally based on a scoresheet. The player can, at any time, ask the arbiter to provide him with a scoresheet, in order to write the moves.
So, if the players get less than 60 minutes for the game including increment calculations then there is no need to record the moves and the first time the player's clock goes under 5 minutes they can stop recording if the increment is less than 30 seconds. I phrase it like this because after the player's clock goes under 5 minutes they can make a few quick moves leaving them with more than 5 minutes. In those circumstances they don't have to start writing again.
In a FIDE rated tournament where the players have 60 minutes or more, e.g. 30+30 there is still no guarantee that they have to write the moves down. This is because FIDE has "holes" in the time controls where it only rates the games of stronger players.
Here is what FIDE Rating Regulations effective from 1 January 2022 have to say:
1. Rate of Play
1.1 For a game to be rated each player must at the start of the tournament have the following minimum periods in which to complete all the moves, assuming the game lasts 60 moves.
Where at least one of the players in the game has a rating of 2400 or higher, each player must have a minimum of 120 minutes.
Where at least one of the players in the game has a rating 1800 or higher, each player must have a minimum of 90 minutes.
Where both of the players in the game are rated below 1800, each player must have a minimum of 60 minutes.
Hence for a FIDE rated competitions both 120+30 and 90+30 games would be FIDE rated and writing the moves required regardless of the strength of the players. However for 45+45 only games where both players were rated below 2400 would be rated and only those players required to write the moves.
The 2400 rating restriction is new this year. It used to be 2200. I kind of understood the 2200 restriction. It was because players above that level could compete for IM title norms and the restriction meant that the games they played that could qualify for a norm had to be at least 2 hours. Extending the range to 2400 means that IM norms could be gained by playing 90 minute games but GM norms would require 2 hour games.
For the lower restrictions for >=1800 and <1800 it doesn't make much sense to me. The USCF rules seem more logical.