Aerith is her correct name.

But it's Aeris for short.

Square are consistently screwing up pretty simple names though, so I can understand not wanting to trust them when they say it in things like Crisis Core. After the debacle of 'Tidus' and 'Zidane' I won't be surprised by anything. Expect to see a remake AVALANCHE team consisting of Bye-gs, Wedd-geh and Jee-see.