The official should not grant a player's request for a time-out until after an injured player has been replaced when a substitute is available.

Does this mean that only the coach may request a time-out if he/she wants to keep the player in the game? Can a team captain or another player instructed by the coach call time-out?

It means they want you to make sure the sub gets reported and becomes a legal player by entering the playing area before granting a timeout. Gives the opposing coach a fair look at who he needs to plan for during his timeout. After the sub is in head coach or a player can request the timeout just like normal.

Really all that it is telling you is not to grant a TO during the 30 second replacement period. This rule was in the book prior to the one that allows a player to remain in the game with a TO. Used to be that once you had blood or injury, you were required to sub and the official had to get the sub before granting a time-out.

The time-out to remain in the game could be viewed as an exception to this older rule.
Since we know that it is newer and should supercede.