The term "police action" was in fact more or less set by accident due to an offhand comment by Truman, and then was used immediately after as a derogatory phrase. In other words, it was intended almost from the beginning to mock the military action, as opposed to being something official that was formed and then pulled apart later. Many other terms were used, and those who really wanted to mock used a term like "Truman's Folly".
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Police action referring to "military action without an official war" was not really in use pre-WWII; the British in India did use it in reference to a large group of officers in response to riots, but given the British were already there in force, it still doesn't quite match the modern cavalcade of undeclared wars that involved forays into foreign countries. Post-WWII, though, you had the name given to the annexation of Hyderabad State by India in 1948 (aka Operation Polo) and the Dutch applied it to operations in Indonesia starting in 1947 (they claimed the lives of Dutch nationals were at risk).
People did pick up on the odd sense of the "police action" term; quoting Life magazine from 1949:
Police Action or War? Guerrillas fight back against Dutch in Indonesia
Last week news pictures depicting the Dutch effort to suppress the 3-year-old Republic of Indonesia reached the U.S. With the pictures came news that the "police action" was really a small war, being fought in the forests and mountains and along the muddy streams of Java and Sumatra.
Part of what was at contention here was the formation of the United Nations in 1945. It was unknown at the time if the UN would be as toothless as the League of Nations, but still, the designation of an act as military became political, particularly with Chapter VII of the Charter, which discusses the Security Council and when military action of one country against another is justified.
Article 39
The Security Council shall determine the existence of any threat to the peace, breach of the peace, or act of aggression and shall make recommendations, or decide what measures shall be taken in accordance with Articles 41 and 42, to maintain or restore international peace and security.
At no point are the words "police action" used; it uses terms like "combined international enforcement action". The point is that the United Nations was a peacekeeping group, the world just went through a devastating war, and the whole point of the design was a hopeful turning away from war. This meant a general positioning position of military force as defensive, which made terminology awkward when this notion was expanded. This tie-in is important since the Korean War was not solely a US action: it had support from the UN, both militarily and diplomatically.
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The actual launch of war was, for the United States, June 25, 9 am Korean time, with the US ambassador to Korea informing the State Department of an invasion, and the same day, the UN Security Council adopted a resolution condemning the act:
Noting with grave concern the armed attack on the Republic of Korea by forces from North Korea,
Determines that this action constitutes a breach of the peace; and
Calls for the immediate cessation of hostilities;
Calls upon the authorities in North Korea to withdraw forthwith their armed forces to the 38th parallel
Engagements also began the same day; while American naval vessels weren't yet in position, North Korean fighters attacked two airfields (Seoul and Kimpo) destroying a C-64 and causing fighter planes to be scrambled; the 8th Fighter Group did an attack of a convoy at 1700 hours. Civilian personnel started to be evacuated.
On June 26th, Truman was still hoping that the ROK will be able to hold without additional ground troops, and he authorized General MacArthur to use the air force and navy.
On June 27th, the UN adopted a new resolution with the additional call to "furnish assistance" as "may be necessary to repel the armed attack". The air war continued, as North Korean planes again make an approach on two airfields (Kimpo and Suwon) with American planes fighting back, downing 7 aircraft in the process.
On June 28th, the British Far East naval commander put his force under control of Admiral Turner Joy (who later was involved with the very messy negotiations leading to armistice).
I mention all this because it's important to note the situation was fast-moving, tied in with the UN (who seemed to be moving towards invoking their nascent powers), had already invoked allied power (with the British naval forces) and had already involved air battles, with the first US plane being destroyed from the surprise of the attack.
Official announcements went for the non-committal, the original statement being "communism has passed beyond the use of subversion to conquer independent nations and will now use armed invasion and war." On the 28th (the same day the British handed over naval command), Truman went before the American Newspaper Guild (a pre-planned speech, that is, on his schedule before Korea happened); four radio networks came, hoping for a statement about Korea, but none was forthcoming.
Truman gave a press conference the next day (the 29th) and waffled in terms of specifics (again: the situation was fast-moving and the government wasn't sure how everything would land). He evaded military strategy questions, but one question came that he answered:
Mr. President, everybody is asking in this country, are we or are we not at war.
To this he responded: "We are not at war."
A second reporter then followed with if it could be called "a police action under the UN". Truman fatefully replied:
Yes, that is exactly what it amounts to.
Truman was, at this point in time, essentially right. The UN had every indication of supporting South Korea; the attack illegal and directly challenged the prior arrangements; the US had not deployed any ground forces and Truman still hoped the air and navy support would be sufficient.
Unfortunately, this is the name that stuck most firmly (although many others were bandied about, like "limited war", "unreal war", and "MacArthur's War").
The religious publication The Defender under the heading Mr. Truman's War discussed a representative (Frederic R. Coudert) who introduced a bill reaffirming Congress's right to declare war, railing about how Truman "deliberately ignored this right when he threw United States forces into the bloody police-action war in Korea."
General MacArthur was quizzed later on the term on the floor of Congress:
Q: General MacArthur, without prejudice to anyone, I merely want to draw your attention to the fact that for months and months the American people was advised that it was a police action, which was going on in Korea. I was one among many who did not understand it.
A: To me, it was war from the day I got there. The term of police action may have been evolved in the concepts that it was an endeavor to clear and police all of Korea. I don't think the term could have been evolved with reference to the actual type of fighting and battle that was going on.
June 30, the day after the "police action" comment, was the day that ground troops were ordered. Truman informed a group of Congress, and while there was one objector (Sen. Kenneth Wherry, of course a Republican) Truman asserted
I just had to act as commander-in-chief, and I did.
It is possible if Congress had bitten back at this moment they could have made a fuss, but objections at this early stage were muted and it was noted (by a Republican) that this action followed the UN Charter so a declaration was not actually needed.
Truman did discuss on July 3 bring up making a joint resolution for war, but there was concern of lengthily debate and stalling, and given the prior tacit approval (and explicit approval through spending bills), it was decided the resolution wasn't needed. Besides, July 4 was when Congress left on vacation.
Hence, "police action" (and many variants) became the standard, and the phrase that got remembered after the war.
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Lorence, J. J. (2009). Selling the Korean War: Propaganda, Politics, and Public Opinion in the United States, 1950–1953. Taylor & Francis.