That rule only applies if an injury time out has been called. An injury does not itself make that rule apply, only an injury CALL does. Where it says ‘the’ injury instead of ‘an’ injury, it is intended to refer to the injury that has been called under 19.1.1.
Note also 19.1.6, where an injury called as soon as the player is able to call it is considered to have been called when the injury happened. This means that I might have dropped the disc before an injury CALL, but could still get it back as the call is effectively backdated to when the injury occurred.