This is called focus and recompose. Note the position of the focus point in the bottom left picture - it is between the two people. If you tried to focus there, the camera would focus on the background, throwing the subjects out of focus. So you focus on one subject, then recompose so both subjects are now in the frame. You only have to do this in similar cases where the focus point does not cover your subject. If the subject is under the AF point, just go ahead and shoot.

Pre-focusing is a different matter entirely. What the manual is saying is, AF doesn't work very well under certain conditions, so manually set the focus instead. Sometimes this will mean truly pre-focusing, ie focusing even though your subject isn't present: fireworks are good example of this. You manually focus at the distance you think the fireworks will be (using a relatively small aperture to increase depth of field, and thus your chances of getting the correct focus), then when the fireworks pop up into the frame, you shoot.

Sometimes it will simply mean manually focusing on your subject because the AF system won't be able to acquire focus, as can be the case with dark hair or glass. Pre-focusing does not imply focus and recompose.