Why do political realists and liberals conflict in how they understand war?Under which of these theories are we more likely to have war?

"Realists" believe national governments are best understood as entities that act in their own self-interest. The internal composition of the government is unimportant, as are the particulars of its political institutions. Realists are above all concerned about the balance of power. They believe war is least likely when power is balanced, and most likely when there is one country that has overwhelming power and can therefore expect to win any conflict.

Liberals believe that national governments are best understood by the principles that define their laws; they believe that democratic countries are inherently more peaceful because most people desire peace. Liberals are not concerned about the balance of power, but instead the composition of national governments; they believe the best way to avoid war is to make every nation into a democracy.

Both theories are trying to avoid war, so whichever theory is right would result in the least war if it is properly followed. If realism is right and we follow liberalism, a powerful enemy could emerge and destroy us while we were undefended because we were so naive. If liberalism is right and we follow realism, nations will start internal wars (that could potentially involve other nations) because of uneven power structures, conquering other countries for no reason and slaughtering thousands needlessly.

Empirically, who is right? Well, neither is completely wrong nor completely right, but the liberals are generally a lot closer. The world is currently the most peaceful it has ever been, and this appears to be because of the economic, technological, and military hegemony of the United States. This makes sense on a liberal theory: a democratic country is perceived by much of the world as "in charge," so there is less war. This makes absolutely no sense on a realist theory, however; the hegemony is supposed to conquer whatever other countries it wants because that would be in its self-interest. According to realists, it's a bipolar world like the Cold War that's supposed to be maximally peaceful.

To be fair, a lot of realists try to redefine "self-interest" so that the US can try to spread its markets or its values around the world, which it indeed does seem to do. At that point, though, we really lose the distinction between realism and liberalism; liberals always said that democratic countries can and should want to spread their values around the world by peaceful means. What made realists distinctive was their claim that much more narrowly defined interests like acquiring territory and natural resources were supposed to define the behavior of a nation, regardless of its internal structure. That's simply false. Nations do have particular values that they try to spread around the world, and it makes a difference whether those values are democratic or authoritarian.

We do still maintain a certain realist stance in some respects—the strongest military in the world, various diplomatic, military, and espionage compromises that advance national security while propping up regimes like the one in Saudi Arabia—but on the whole, liberalism has been the policy of the United States and it seems to be working.