Looks like you should get it if those months are from 2012- 13, but why are you complaining about this now? I mean even if it is a mistake it might be to late for them to change your account? You should of complained about this a year ago.
Legacy people don't get the Coeurl mount, you only get that with 1.0 CE or ARR CE
Edit-------------
I can see they most likely messed up your account, cause you say you first joined in Sept 3, 2010, (which is weird cause the CE of 1.0 came out a couple days before the standard 1.0 and CE came out on Sept. 22, 2010). And under your avatar it says you joined on August 2013.
Edit #2!------------------
I think I may know what the problem is, From what I know and have seen many people do, you probably had 2 FFXIV accounts, your 1.0 legacy account. And somehow people made a second account for when alpha/beta started and they screwed up and put there beta codes on that new account.
If this is true that means that you are playing on your new beta account that you would of indeed created just this month. And you are not using your 1.0 account. In which case all you would have to do is see if you can go into your account management and somehow merge or re activate your old 1.0 legacy character? I have seen this happen many times , although sorry I don't know how to fix it but it sounds like this is probably your problem, if you send SE another E-mail i would put that you might of done that to. Cause I think SE knows about this problem but if you get someone at customer service that doesn't really care than he will just say no u don't get it, and that be the end of it.
But this would probably be the most likely cause as #1 it says you joined just this month. And #2 you have absolutely (well you have some but not alot) nothing on that character, and if you are legacy i would expect you to have allot more on a character.