How does one determine when a war is won or lost? what is the purpose of a treaty?

The purpose of a treaty is to officially end the state of war between the hostile parties.

You can win or lose a war in many ways. It depends on the time in history. A nation can surrender after enough losses, give the territories to the winner or the whole country in some cases. But capitulation is not the only way to win a war. You can lose every single part of land a country owns but continue to fight with your allies like Serbia did in World War 1. And Germany surrendered in World War 1 not by losing territory but by realizing that there is no way to win the war. And in World War 2 they were fighting until the end. Even after the capitulation, some soldiers kept fighting.

So to win a war you need the other side to surrender. Casualties, territory, and length of war do not mean victory or defeat, only when one party concedes defeat.

abraham lincoln  (february 12, 1809  – april 15, 1865) was an american statesman, politician, and lawyer who served as the 16th  president of the united states  from 1861 until  his assassination  in april 1865. lincoln led the nation through the  american civil war, its bloodiest war and its greatest moral, constitutional, and political crisis.[2][3]  he preserved the  union,  abolished slavery, strengthened the  federal government, and modernized  the u.s. economy.