How we cite our quotes: (Article.Sentence)
Quote #1
His Britannic Majesty and the United States of America [are] desirous of terminating the war which has unhappily subsisted between the two Countries, and of restoring upon principles of perfect reciprocity, Peace, Friendship, and good Understanding between them. (Intro.1)
The language in the first sentence of the Treaty sets a high benchmark for the peace. Not only is the treaty aiming to be fair; it aims for "perfect reciprocity," and friendship between the two countries that had spent the last couple of years shooting at each other.
Quote #2
All territory, places, and possessions whatsoever taken by either party from the other during the war […] shall be restored without delay and without causing any destruction. (I.3)
The end of this war was like resetting a chessboard. Instead of gaining land or cities, each side had to give them back. This aspect of the Treaty explains why many perceive the War of 1812 as a useless war. What concrete objectives did it accomplish? It's hard to point out anything based on the text of the treaty alone.
Quote #3
In order therefore finally to decide upon these claims it is agreed that they shall be referred to two Commissioners. (IV.2)
This part of the treaty refers to "claims," or disagreements, about the possession of certain islands off the east coast of the United States. Appointing commissioners to work things out was like cutting through a knot. Deliberating the possession of numerous islands during negotiations—or worse, turning things over to the legislatures—would have allowed the war to drag on.
Quote #4
The said Commissioners shall by a declaration or report under their hands and seals decide to which of the two Contracting parties the several Islands aforesaid do respectively belong in conformity with the true intent of the said Treaty of Peace of one thousand seven hundred and eighty three. (IV.5)
This quote shows Britain extending the hand of equality to the United States. Instead of trying to change the treaty of 1783, aka the Treat of Paris, aka the treaty that ended the Revolutionary War, the Brits shrugged their shoulders and let it stand as is. As opposed to, say, demanding New Jersey back.
Quote #5
And His Britannic Majesty and the Government of the United States hereby agree to refer the report or reports of the said Commissioners to some friendly Sovereign or State to be then named for that purpose, and who shall be requested to decide on the differences which may be stated in the said report or reports. (IV.8)
Here the two countries agree on a method of resolving disagreements between the Commissioners themselves: give the final call to another country. Agreeing to appoint an impartial judge or mediator is definitely a gesture in the direction of equality, kind of like hiring a divorce mediator. You hope it never gets that far, but if it does you can at least try to be fair.