How to install a chain that has a pre-fitted pin? My new chain (Shimano 5800 HG600 105) comes with a pin inserted in the outer plate as seen in the picture below.

The Shimano dealer's manual makes no mention of this scenario.

How to install such a chain?
Shimano 11 speed (5800) should be assembled with a Shimano chain connector pin. The pin is pushed through and the end breaks off.  9/10/11 speed chains outer plates are too narrow to reuse pins.  There is literally no margin for error.  You actually break part of the pin when you push it out, as such it won't stay in place if reinserted. A missed shift can easily pry the outer plate off the pin causing the chain to fail.  

I suspect you were sold a used or pre-installed chain, where they pushed the pin partially as a "service" to you, but if you reinsert that pin in all likelihood the chain will fail at some point in the future.


Shimano chain connector pin

UPDATE

The online retailer (guessing chainreaction) may also be selling an OEM (Original Equipment Manufacturer) chain:


  CRC supply OEM chains which have a short rivet pin partially inserted ready for use. Never had a problem since I learnt how to use this rivet. You have to be careful that you get it in the right position first time when using your chain tool, as there is no room for error, when compared to the long pin type of rivet. If you try to adjust the chain pin it soon comes loose.
  It isn't just a removed pin, as these are always too loose to use, as part of the rivet shoulder breaks away during removal.
  Mind you it's always handy to have another way of joining the chain if you don't get the pin right first time.
  I find that you have to just take your time when pushing the pin through the last bit to avoid going too far through.
  -- Forum Thread


I have never personally had access to OEM (equipment purchased by bicycle manufacturers) so I cannot confirm whether or not this is how the chains come.  Either way, the forum thread suggests assembling with these short pins can be difficult (other answers discuss how to do this).  You may wish to purchase a connector pin as this is how Shimano intends home mechanics to assemble chains.

To clarify this short pin may not be the same as a pin that has been pushed out on an assembled chain. But it is hard to verify what the OP has.