The fact that your account says '' Final Fantasy XIV '' just means you added the FF 14 to your account by any key whatsoever ( this could have been a beta key, or the early access code )

My account said '' Final Fantasy XIV Registered 28 July '' ever since I added my post Phase 3 beta key, it has nothing to do with your early access working.

Like people said the way to check is using the launcher, if you have early access you will have a non clickable play button.
If you don't got early access it will say you need an account with Final Fantasy XIV in order to play