Quote 1
"See, Gawain, that you carry out your promise exactly,
And search for me truly, sir, until I am found,
As you have sworn in the hall in the hearing of these knights."
(448 - 450)
The Green Knight reminds Gawain of his knightly duty to keep his promise. The fact that Gawain has sworn to do this in the hearing of all the other knights means that his reputation will be damaged if he fails to carry it out - it’s a matter of knightly honor.
Quote 2
"Everything I ever promised you I shall readily give."
(1970)
This is Lord Bertilak’s response to Gawain’s request for a man to lead him to the Green Chapel. It becomes ironic in light of the ending, when we learn that Sir Bertilak is also the Green Knight. Lord Bertilak might also be referring to his promise to return the stroke Gawain gave him in Arthur’s court one year ago.
Quote 3
"Let us repeat our agreement before going further,
First I entreat you, sir, that what is your name
You shall tell me truly, that I may believe you."
"In good faith," said that virtuous knight, "I am called Gawain,
Who deals you this blow, whatever happens after,
On this day next year to accept another from you
With what weapon you choose, and from no other person
on earth."
(377 - 384)
It’s important to the Green Knight to learn Gawain’s name so that he knows who to hold accountable to their agreement. This provides Gawain with an incentive to keep their agreement, too, because he knows his reputation will be sullied should he default on it. The Knight also insists that Gawain repeat the terms of the agreement, probably so that Gawain can’t claim ignorance as an excuse after the fact.