Is it better to shut down the engine(s) or leave it (them) running when a gear up landing is imminent? Is it better to shut down the engine(s) or leave it (them) running when a gear up landing is imminent?

I noticed that the pilot in this video chose to shut down his engine before touching down during a  gear up emergency landing.  However, in Pilot Operating Handbooks, the recommendations seem to be mixed.  For example, the Cessna 210 POH, Page 3-11 says:


  
  Touchdown -- SLIGHTLY TAIL LOW.
  Mixture - - IDLE CUT-OFF.
  Ignition Switch -- OFF.
  Fuel On-Off Valve -- OFF (pull out).
  


On the other hand, the POH for the Piper Arrow III PA28R-201, Page 3-14 recommends shutting down the engine before touching down:


  3.lSb Gear Up Emergency Landing (3.Sd)
  When committed to a gear up landing, CLOSE the throttle, move the
  mixture to idle cut-off, and shut OFF the ignition, battery master (BATT
  MASTR), and alternator(ALTR) switches.Turn OFF the fuel selector valve.
  Seat belts and shoulder harness should be tightened. Touchdown should
  normally be made at the lowest possible airspeed with full flaps.


What is the reason for this variation?  Is there some advantage to touching down one way vs. the other?

I've have noticed this question:


  How to execute a gear up landing on aircraft with low propeller?


However, I do not believe this question duplicates that one. While consideration of the propeller would be one factor in deciding whether or not to shut down the engine, it certainly would NOT be the only one.
There's two schools of thought. First is the keep the engine on school because keeping the engine on:


Gives you more control of the airplane
Gives you the option to go around 
Keeps the procedure simple and allows the pilot to concentrate on a good touchdown
Keeps the dynamics of the airplane predictable and familiar to the pilot


The shutdown the engine school's view is that shutting the engine down lowers the fire risk by cutting off the fuel and engine ignition spark. 

Personally I'm with the keep the engine on school of thought. The last thing I want to be doing when I'm trying to make a smooth landing is reaching for all sorts of knobs and levers, and having to reach the fuel cutoff valve in the inconvenient places that piper likes to put them, and then learning how the airplane behaves without engine power for the first time. As soon as I touch down I am pulling the mixture, cutting off the fuel and killing the mags and the master switch, but until then I'm going to concentrate on making it the smoothest landing in the history of manned flight.