Why is it called "The Legend of Zelda," anyways?

I thought about this question a while back. Hey, it's what I do.

There are two reasons, from what I can tell, for the name of the series.

1.) The ACTUAL 'Legend of Zelda.'
The Legend of Zelda refers to an old story first told, if I remember correctly, in the instruction manual that came with Zelda II. The story told that long ago, the Hyrule was led by a king. This king had a son and a daughter. The king became ill, and on his deathbed he told his daughter, the Princess Zelda, the location of the hidden Triforce. He died, and the Princess became the only one who knew of its whereabouts. The Prince and the king's adviser both wished to know the location of the Triforce, but Zelda wanted it to remain a secret. In his anger the adviser, actually a powerful sorcerer, cast a spell on the Princess that would keep her in an indefinite sleep. In the process of this spell, the sorcerer died. The Prince was so upset about the fate of his sister that he declared that from that point on, all daughters in the Royal Family would have the name Zelda.

2.) A hero of many names
In each Zelda game, you can name the hero whatever you want (within the space limitations). This is the reason why the game isn't called "The Legend of Link," as it would actually be "The Legend of [whatever name you input]." Obviously Nintendo didn't like this idea, and seeing as the original game had only Zelda and Ganon as in-game, named characters, along with the fact that they likely didn't want to name the series after the antagonist, they went with "The Legend of Zelda."

Those are the reasons I've come up with. What do you think?