How does a water meter affect the continuity of a grounding electrode system when a jumper is present?
The grounding electrode is only required to be 8' and buried 2-1/2' deep for a pipe or rod. The conductor is required to be continuous as you note but you stopped reading prior to the 4 exceptions 250.64.C.1 through 4.  But the pipe is the electrode.
 With many home owners having to replace old galvinised pipe a problem has started when the old pipe is removed and replaced with plastic. I have had to drive new rods where ther was no longer 8' of pipe in the ground. This can cause problems especially if there are no other grounds (driven or Ufer) the most common. If you are concerned about the quality of your ground it is fairly easy to drive a new rod close to your panel and run a #6 copper conductor to the service. Many older homes the water main was the only ground. Adding a driven rod or pipe can only help if located close to the service.