The title basically says what I really want to know.
I really love Ahri and was hoping to use her to carry myself out of bronze. But I never realized if i was playing Assassin or Mage Ahri, if it helps, i usually rush DFG and charm away & annihilate single targets utilizing my magical charm.
Mage? Regardless of what RIOT says in game, Ahri has been and always will be an Assassin.
There's no difference. Ahri is a champion with only 1 build path, and that's assassin.
If you're behind, go defensive and build 
vs ad and 
vs ap
If you're ahead and want huge mid game burst build 
if you find you are running low on mana build 
If you find you need early game power and are ahead build
On AP assassins, 
IS A HUGE NECCESSITY.
NOBODY IN THEIR RIGHT MIND WILL TELL YOU OTHERWISE.
If you make a mistake or go out of position, will give you ample time for your team to come try to help and you think of a clever way to escape.
I play Ahri like a mage, because I play really safely and wait for teamfights to happen, where I actually do well. However, I still build her the same way as most people should, with DFG with Zhonya's/Deathcap etc.
I never seem to get a 100-0 kill, someone else always seems to take the last hit away from me after doing like 95% of their HP. :
