Exactly what does the last part of the first bullet point mean?
"... unless the ball rebounds from the goalkeeper or the goalkeeper has made a save."
I have no idea what they are trying to say.
Ok. So breaking it down, the clause says "The goalkeeper is considered to be in control of the ball with the hands... by touching it (the ball) with any part of the hands or arms, except if the ball rebounds from the goalkeeper or the goalkeeper had made a save."
So the goalkeeper has control of the ball merely by touching the ball with his hands or arms, other than if the ball rebounds from him. This raises the question when the ball could be touched by the goalkeeper's hands or arms but the touch is not a rebound. The scenarios I can think of are:
1. The goalkeeper is holding the ball with one or both hands.
2. The goalkeeper has the ball between his hand and any other surface, such as the pitch, the goalpost, another player (but if the player moves away, the ball could then be dropped and the goalkeeper would no longer have in control of it).
3. The goalkeeper could be lying on the ground with the ball inadvertently stationary, yet touching any part of his arm or hand.
If the ball touches a goalkeeper's hands or arm in any other than these scenarios, it would constitute a rebound.
Even in the third scenario above, whilst the ball is in contact with the goalkeeper's hands or arm, no other player would be allowed to play the ball. The goalkeeper has six seconds (often interpreted by referees as anything between 6 and 20 seconds) by which time the ball must be released back into play.
This is my personal interpretation based on the written LOTG and logical deduction. Official advice or clarification given to referees could well differ.