I have seen a lot of problems/confusion with this lately.
Specializations are end-game content, you must hit rank 50 to use them, regardless of your edition.
For any edition of the game, standard, limited, console, whatever… you do not get specializations day 1. (Unless you hit rank 50 on the first day…).
When you hit rank 50 of the standard edition, you have access to two specializations. The rest of them will be gradually unlocked by 343i depending on how many people are hitting the right rank.
When you hit rank 50 of the limited edition, you have access to all 8 specializations. (Remember, you can only rank up in one at a time). You do not get all 8 at launch, you must get to rank 50, then you have 6 more than the standard users would normally have at first.
Specializations are end-game content. Hope this helped.