I believe that two new primals coming to FFXIV in the Heavensward expansion will be Knights of the Round and Midgarsormr. (THANK YOU 2.4!)
Yoshida announced during the opening keynote address of Fan Fest that 3.0 would introduce two new plot points to FFXIV: 1) The 1000 year Dragonsong War between Ishgard and the dragons of Dravnavia, and 2) The Wyrmking vs Thordan & his Knights Twleve. He also announced that there would be new primals (ssss was the sound made, like PLURAL) in 3.0. In addition, Yoshida stated that the logo art contained hints about 3.0's plot.
I believe the art is showing us more than a battle between dragoons and a dragon. I believe it's showing us either Ishgard vs the Wyrmking (Midgardsormr) or KotR vs Midgardsormr.
I think it's safe to say from the obvious hinting in Yoshida's keynote that KotR will be in the expansion. Thus far the old Final Fantasy summons have been either turned into primals (Ifrit, Garuda, etc.) or boss battles (Diabolos, Siren, Fenrir in Snowcloak for 2.4) Since "the Knights Twelve" is a major plot point, I'll go with primal for Knights of the Round. I'm currently not worried about who will summon KotR and I can't really further speculate without more teasers, maybe something from London or Toyko Fan Festivals in upcoming weeks.
So let's check Midgarsormr's lore: The official 1.0 lore video The Waning of the Sixth Sun says Midgardsormr has been asleep for centuries and was with dragons in battle versus the empire. Gamer Escape at http://ffxiv.gamerescape.com/wiki/Midgardsormr tells us that "was a great dragon that lived in Mor Dhona from primeval times until his death in 1562 (Sixth Astral Era) in the Battle of Silvertear Skies at the hands of the Garlean Empire's massive airship, Agrius." and also "When the Empire invaded Mor Dhona, knowing it to possess Eorzea's greatest concentration of aether, they incurred the wrath of Midgardsormr, who burst forth from the water to fulfill his purpose as its guardian. Despite fighting "with divine strength and purpose," he, and the dragons that came from Dravania to fight alongside one they revered as a king, could not overpower the might of the Empire's armada."
One can see how at a place of great concentration of aether and worshipers that a primal could appear. Bahamut has been called the Elder Primal of Dravnavia's dragons but Elder could mean one over other lesser primals. I also haven't found much to substantiate that and (without spoilers from Binding, Second, or the upcoming Third Coil) just look at the body of Bahamut from art and the 2.4 trailer: http://youtu.be/QbiFK4tcz9w?t=7m8s. Bahamut's chest seems a bit too big for the logo. Maybe we'll know more once 2.4 is out and Final Coil is beaten.
EDIT: Seems like 2.4 has told us more about Midgardsormr. A spoiler from very early in 2.4's quest line is below. You are warned.
Here's the logos and art side by side. You be the judge.
**SPOILER ALERT from Patch 2.4** (very very early in the quest line)
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