I'd argue that the fact that there is a supernatural element clinches the deal.  Death/Fate personifications are a common element amongst even some of the very earliest horror stories.  Poe's stories revolved around inevitable threats, as did those of Lovecraft.  Even Stoker and Shelley personify inevitability as the thrust of their stories.

Now, I'm not putting Final Destination on that level.  Basically, I think the execution of those stories will prevent them from ever being essential classics... but they are definitely horsing around with the elements of classic horror.

Do I claim to be a TRUE horror fan?  Maybe.  That's a fair question - but were I a true horror fan, my firm position on the subject would be that if death itself is trying to kill you, you must be in a horror story.