Well, the Dalish codex isn't necessarily "wrong". It was already super abbreviated. All it says is "First came missionaries. Then Templars. Then exile." You don't exactly have to reach World of Thedas to know there was probably a whole bunch of other stuff that also happened in between those 3 things as well. And as for the Chantry vs Orlais being behind it, elves could very easily perceive the two as being effectively the same thing. They know Chantry missionaries were sent first in and that the conflict had a religious bent by the end with the Divine calling for a Holy War and their religion being outlawed. It makes it very easy to perceive that the their religion was the true motive the humans had for fighting the war.
And while Orlais and the Chantry are not the same entity, they're related closely enough that the elves could still be partially right.
I also don't think it's fair to dismiss the opening skirmishes as the start of the conflict. Technically we also don't know that the elves' aggression against Red Crossing was a deliberate attempt to start a war either. It could have just been a raid intended to really just stick it to some nearby shems after the last border skirmish.