Yeah, is there a really "official" story, and WHO and WHERE changed it anyway ?
In the original Japanese, her name is Earisu. The su can be transliterated as both 's' and 'th'. The intended spelling is generally accepted as Aerith, as the official Japanese stuff usually uses it, and the game's designers have mentioned that Aerith was intended to be a near anagram of Earth. So neither is really incorrect, but Aerith is more official.