I'd been hoping you'd get a specialist on the war from an American perspective to answer this, but unfortunately it looks like that may not happen. Apologies -- this won't be as thorough an answer as I'd like, but I hope it helps a little!

The Korean War was framed very differently from World War II in domestic U.S. politics. It was seen mostly as an opportunity for the still very new United Nations (~5 years at the time of Kim il-Sung's invasion of South Korea) to fulfill the purpose of its charter with a very straightforward conflict. As the West saw it, South Korea was being invaded: South Korea did not wish to be invaded: South Korea had the right to expect assistance from the international community to preserve its territorial integrity and protect itself against the forceful imposition of a government whose authority it did not recognize.

The matter went immediately to the U.N. Security Council and resulted in only the second SC resolution of 1950, condemning the invasion by North Korea and recommending that U.N. member states provide assistance to South Korea. SC resolutions are a lot more common today than they were in the early days of the U.N., and that was considered a more serious pronouncement than we'd perhaps be inclined to think of it today. While the U.S. provided the overwhelming majority of soldiers and weapons for the U.N. campaign, it was framed as doing so as part of a wider Western effort against the relentless spread of a hateful ideology (communism).

In truth, Kim's sudden invasion of the South played a significant role in how the American public came to see communism for the rest of the Cold War. Communism was seen as an aggressively expansionist belief system that relied, not necessarily on popular support, but on the trickery and violence of repressive regimes that were backed by a united cadre of nations within the Soviet Union's sphere of influence.

It's perhaps most helpful to observe that Kim's invasion was firmly within the Red Scare period of mid-century America; it happened four months after Joseph McCarthy shot to national prominence for claiming that he had a list of Communists who were secretly working in the U.S. government, and two months before Julius and Ethel Rosenberg were indicted by a federal grand jury for supposedly passing nuclear secrets to the Soviets. Kim didn't help matters by his appearance on Korean radio just a week prior to the invasion, reiterating his commitment to a peaceful reunification of the peninsula while moving soldiers and weapons to the border. So in terms of "selling" the conflict to the American public, part of this was certainly influenced by the idealism of the early United Nations, but I would argue that it's better seen as evidence of Americans' growing hatred and fear of communism.

However, we should also point out that Korea itself wasn't a prominent part of U.S. public consciousness beforehand (or really even during or after -- there's a reason that it's so frequently called the "Forgotten War"). The war didn't require the mass numbers of troops or economic reorganization that World War II had required, the public knew little of Korea anyway, the country didn't have the high profile of WWII's vanquished enemies and their needs, and the war had no serious impact on day-to-day life in the United States. Even the government was in the midst of removing soldiers and equipment from the peninsula in reciprocity for the Soviet draw-down; there was little public appetite for another expensive and diplomatically sensitive occupation.

So -- if this makes any sense -- the Korean War was much like World War II in that it was seen as necessitating U.S. action as part of a multilateral group to fight the spread of evil, but wholly unlike World War II in that it didn't reshape American society to any serious degree.

I found an interview with Melinda Pash on the cultural background to the Korean War and why it's so frequently "forgotten" in America, and I think this might help a lot with your question.