I don't think the OP has said or implied that at all...
My understanding is that the POD is an actual declaration of war between the DPRK and the US. Officially, there was no war, just a 'United Nations Police Action', which has influenced successive conflicts. No war with North Vietnam or Panama or Iraq, which resulted in the War Powers Act of 1973 which put approval of medium-term (90+ day committments) in the hands of Congress.
The question is, then, what effect would having to approve every war-like conflict, and designate a country to be defeated and occupied like World War 1 and 2, Mexico and (technically) Spain, have on the progress of Korea's war, Vietnam and other committments, and on the new asymmetric conflicts with non-state entities. It may also involve a question of formulating International Law with regard to wars of other nations - I don't think there were formal declarations of war, for example, in the Holy Lands, African multinational efforts to remove Idi Amin, Liberian, Sierraleonian and other despots, etc.
For my view, I don't think anything much would have changed in Korea. China (backed by the Soviets) was too powerful to fully war with, so any move to declare a state of war would have had the same force behind it in a formalized or non-formalized system.
The conduct of the war would have been the same - there was widespread and multinational commitment. Ike's nuclear threat would have had the same force and thus effect. Stalin's death occurs on time, his views on the Korean War would not change because of some words.
The main change stemming from an actual Declaration would be that the United Nations would have less of a look-in. The view of Americans and others would be that this was a war between Korea and the US (and allied countries) rather than the birth of the United Nations. The emphasis, that is, would be on the first part and not the second. The UN itself would, I think, remain as it has been, involved in debate, a council of nations, tool for peacekeeping, etc.
The necessity of naming North Vietnam as an official enemy in order to bomb it in 1965 would, I think, present more problems. With the example of North Korea in their minds, the Soviets and Chinese, while not wishing to risk their own lives for Vietnam, would consider it a matter of honor and credibility to support a communist state under attack from a treaty-breaking capitalist bully (as they would see it). Given the ouster of Hruscov for lacking firmness, I can't exclude the possibility of severe tensions and some small-scale 'incidents' between the Soviets and Americans. Domestic dissent would, as a result, be looked at with less favor. It might also butterfly some of the antiwar movement's legitimacy. Or it may force the US to abandon South Vietnam once the Congress and people are asked to publicly support attacking a small country far away in order to defend an unwillingness to abide by the Geneva Accords.
Even if nothing else changes up to 1973 (and there will be changes), the War Powers Act is butterflied because Congress doesn't have a history of Presidential private wars being waged to refer to. Having been asked to approve or deny wars up to that point, there is no need or desire for the WPA.
And butterflies, and effects, continue....