Nobody used the name, but Moiraine recognized the banner of LTT by the creature on it for what it was the second she saw it.

Whether the creature is known as Dragon or not is immaterial. In educated circles (and Hawkwing was such, and knew the prophecies well), people know the red-and-gold creature like a serpent with legs was the sigil of Lews Therin in the War of Shadow. The Kinslayer - not at all a person you want to be associated with, unless you are a false dragon. So the question remains the same: why would Arthur Hawking put the symbol of the Kinslayer on his scabbard? He spent several years actually bringing down a false dragon who wrecked half the world before he was stopped, then the aftermath of this capture brought war and woes for years before Hawkwing brought it back to peace, conquering or getting allegiance from everyone - what could have possessed the man to use the symbol of the Kinslayer himself?

I don't see a logical reason myself. If anything Hawkwing is the antithesis of a 'dragonsworn'.

Now, if that scabbard had a the golden hawk on blue, we'd be getting somewhere. But it hasn't.

Half the world is under Rand's protection now, and half the world is under Tuon, who has organized her half quickly and efficiently. The two should be 'as one' before the Last Battle. This is perhaps what the distorted story from the fourth age is about, not Rand having Justice, Rand and the Seanchan as protectors of the Light. Or, for all we know, the legends of the Dragon and Hawkwing with justice have been merged by that point - notice it has already distorted LTT's titles - not Prince of Dawn/Lord of the Morning, but Lord of the Dawn, and how Hawkwing's sword Justice is transformed into the sword of justice.