This might’ve been an older traditional rule and it may or may not be still in practice. Though when you look at the records of Japanese games, often they do stop recording the game record just before the dame/teire phase, so maybe subconsciously they do have a phase like that 
It does technically show up in the EGF tournament rules
Finite thinking time under Territory Scoring The following applies under Territory Scoring. It is not compatible with the Ing Overtime System.The alternation consists of the competitive phase followed by the neutral phase. During the competitive phase, one or both players moving next can make a play to improve the score or to fill a basic endgame ko. During the neutral phase, neither player can do so because only dame and teire, if any, are left.
If the first two successive passes occur prematurely during the competitive phase, then the clock is neutralised, each player’s time is set to exactly 1 minute, and the clock is restarted for resumed alternation.
Until two successive passes during the neutral phase, every legal play is considered sportsmanlike.
During the neutral phase, a player has to pass if his opponent has just passed. Then on neutralised time, more dame and teire may be filled quickly in continued alternation.
I mean when we play in person, sometimes you do have people passing before dame, or stopping the clock to fill dame. I think you might especially do it in a blitz game, to be fair to your opponent. I don’t think it’s very sportsmanlike to force the opponent to fill dame on the clock, or keep passing because it would be quicker.
If this is still the confusion, you can think of it as like a good habit if you want to avoid mistakes. You can think of it as a way of removing ko threats, in cases there’s the threat of a ko (it can happen from mistakes, it can show up in the dame filling, there can be a ten thousand year ko or some funny situations where a ko isn’t resolved until the very last move).
On the other hand you can just think of it like a kind of implied move order.
Ordinary endgame: where both sides could gain/lose a point. Black will probably get to play here first, but white in theory could push in once before defending.
Teire: Nobody really gains points, but white will eventually owe a move inside to fix a weakness, shape etc.
Dame: In Japanese rules, nobody gains any points, there’s no need to respond to the move even.
Typically endgame moves that can change the score will be played before teire, and teire will be fixed before the last dame.
There’s exceptions, you might force someone to fix the teire as a ko threat, you might squeeze it in if the follow-up is big enough that someone can’t make a gain by ignoring it, but you don’t lose anything typically by leaving it until much later.
As a rule of thumb, should I do it or not? As you can see, the beginner’s book is trying to ingrain the habit of doing it (make the defensive moves before dames). I’m sure there are situations where this is incorrect (nothing is absolute in Go), but just a point of departure, something to be aware that you’re probably right, before acquiring better reading skills …
If making a move that gains points during the dame phase is considered poor sportsmanship, I wonder what a championship would be like if it were like a poker table, with a player teasing the other (although I’m not sure how far that’s still allowed) 