Does the nervous system have "routers"?
The second model is the one. No tags, flags, or routers, just a 1-to-1 system. 

A striking example of this is are the different effects of spinal cord injury, which depend on the place where the injury occurs. 

  

And I quote from the apperalyzed website:


  Spinal Cord Injury Overview
  
  When a person suffers a spinal cord injury, information traveling
  along the spinal nerves below the level of injury will be either
  completely or partially cut off from the brain, resulting in
  Quadriplegia (Tetraplegia) or Paraplegia.
  
  [...] The brain will still be trying to send messages downwards to the
  muscles in the body, known as motor messages. These messages however,
  will be blocked by the damage in the spinal cord at the level of
  injury. Nerves joining the spinal cord above the level of injury will
  be unaffected and continue to work as normal.
  
  Quadriplegia / Tetraplegia [occurs] when a person has a spinal cord injury above 
  the first thoracic vertebra. Paralysis usually affects
  the cervical spinal nerves resulting in paralysis of all four limbs.
  [...]
  Paraplegia [occurs] when the level of spinal cord injury [is] below the first thoracic spinal nerve [and leads to] paralysis of the legs and abdomen up to the nipple line, [but] paraplegics have full use of their arms and hands


This example shows that there are strict one-to-one connections (2nd model) and that messages are not simply flagged and routed to their destination via the shortest or quickest route available among a bunch of redundant, ill-defined pathways (1st model). Hence, there is only one way from the brain to the periphery; once severed the message cannot be simple re-routed via parallel lines.   

Another example is the make-up of the sensory-motor cortex:



The picture shows that the motor cortex (muscle control/motor output) as well the sensory cortex (sensory input) contain a complete image of the body. Each part of the motor cortex and sensory cortex is devoted to the control or sense of one specific part of the body, respectively.

Nerves are indeed typically bundles of 1-to-1 fibers. For example, the optic nerve contains fibers that carry info from one, and only one well-defined and localized pixel from the retina to the brain.

Principles of Neuroscience by Kandel and Schwartz is a great book with a wealth of info.