How we cite our quotes: (Article.Sentence)
Quote #1
It having pleased the Divine Providence to dispose the hearts of the most serene and most potent Prince George the Third, by the grace of God, king of Great Britain, France, and Ireland, defender of the faith, duke of Brunswick and Lunebourg, arch-treasurer and prince elector of the Holy Roman Empire etc., and of the United States of America, to forget all past misunderstandings and differences that have unhappily interrupted the good correspondence and friendship which they mutually wish to restore, and to establish such a beneficial and satisfactory intercourse , between the two countries upon the ground of reciprocal advantages and mutual convenience as may promote and secure to both perpetual peace and harmony. (Intro.2)
This is as close to a thesis statement as the treaty gets. It specifically calls out "friendship" as being part of the goal of the treaty itself. While it's easy to dismiss this as making nice for diplomatic purposes, it's important to remember that making nice for diplomatic purposes has a—wait for it—diplomatic purpose.
Specifically, of helping turn enemies into allies. Whether or not Britain meant that they wanted the U.S. as their BFF is nearly irrelevant. They left the door open, and that alone is a good enough reason to do it.
Quote #2
It is agreed that the people of the United States shall continue to enjoy unmolested the right to take fish of every kind on the Grand Bank and on all the other banks of Newfoundland, also in the Gulf of Saint Lawrence and at all other places in the sea, where the inhabitants of both countries used at any time heretofore to fish. (3.1)
This part of the treaty governs fishing rights, which were more important then than they are now. In it, the British are assuring the U.S. that they basically have all the fishing rights they want, which is huge. It's pretty friendly.
Quote #3
That there shall be no future confiscations made nor any prosecutions commenced against any person or persons for, or by reason of, the part which he or they may have taken in the present war, and that no person shall on that account suffer any future loss or damage, either in his person, liberty, or property; and that those who may be in confinement on such charges at the time of the ratification of the treaty in America shall be immediately set at liberty, and the prosecutions so commenced be discontinued. (6.1)
The British king is looking after the Loyalists here. This is an important bit of friendship. These people never turned their backs on him, so he's asking they be treated well. It's a stand-up move, and in doing so, he makes a relationship with the U.S. easier.