Yeah, on the Outer Planes everybody would consider the Arborean Eladrin to be "the" Eladrin, and only on the Feywild or the Prime would folks think of the Feywild Eladrin as "Eladrin". I can imagine some Arborean Eladrin (and some other folks on the Planes) flat out refusing to refer to the Feywild variety as Eladrin, and even using some terms of abuse I've heard for them such as "Blink Elves"! Of course, not EVERY Arborean Eladrin has to take that attitude towards the Feywild Eladrin; after all, in my story, they used to be close allies... and maybe could be again, if one side or the other were to apologize. Problem is, each side is sure it's right; the Arboreans, that they were justified in banishing their admirers, and the Feywilders, that they didn't do anything wrong in openly opposing evil on the Prime... so what if there's a bunch of stuffy rules that say otherwise?
Of course, that's not the only way you could reconcile the matter. Maybe both types of Eladrin are still bosom buddies, and the bunch on the Feywild moved there for other reasons...