The answer here is...it's complicated. I think generally the issue will come down to semantics, like arguing whether extensive foreign intervention (to the point that the majority of the war effort eventually fell upon the US and PRC) means it was not a civil war.

To give a brief overview, Korea was a single country prior to its annexation by Japan in 1910. With Japan's defeat in the Second World War, a group of exiles (of varying political opinions) who had formed the Provisional Government of the Republic of Korea attempted to stamp their authority on the peninsula and effect Korea's transition back into a free country. However, the reality went differently, as the USSR had declared war on Japan and invaded Manchuria, swiftly advancing into Korea as well. In response, the US hurriedly asked to divide the Korean peninsula at the 38th Parallel, and so the country was split by the actions of two foreign powers. Initially the superpowers spoke of reestablishing an independent, unified Korea, yet it played out differently as the Cold War developed and they shifted their aims towards containing each other's influence in east Asia.

I'll keep the focus here on the Korean side of things since it's the most pertinent. Due to the occupying governments in the two zones, different groups rose to prominence and defined the ideological outlook of their half of the country. In the North, the Soviets placed Kim Il-Sung at the head of a communist government which encompassed Korean communists from a range of backgrounds, who consolidated power rather quickly despite their differences (many would be purged after the armistice). Meanwhile, in the South there was a political struggle between rival factions of conservatives led by the likes of Kim Ku, Syngman Rhee etc. This only died down somewhat after the 1948 election put Syngman Rhee in power, but never fully went away before the 1950 invasion.

To break the international deadlock, the US referred the problem of Korea to the UN in 1947, and under its supervision all of Korea was to vote for a single legislature to which the superpowers would transfer control. In the end the USSR called foul and failed to cooperate, so the election only occurred in the South. On this basis, the Republic of Korea was formed. The North then held its own, separate, election and proclaimed itself the Democratic People's Republic of Korea. The most crucial point (in my view) to note for your question is that both governments claimed jurisdiction over the entire peninsula as the successor to the former state of Joseon and viewed the other as the puppet government of an occupying invader. Both also explicitly stated their intention to reunify the peninsula by force if need be (which led to the US deliberately making the ROK Army poorly equipped, to prevent them from starting a war which the US did not want).

So back to your question - was the Korean War a civil war? As I said before, it comes down to semantics. The Cambridge Online Dictionary defines a civil war as:

a war fought by different groups of people living in the same country

That is very broad, and points you to the next semantic issue by needing a firm definition of a 'country'. So was Korea a country on a cultural or political basis, and if so how did the splitting of the country factor into that?

As Korea was formerly a united country, and that as both states claimed to be the successor to that country and claimed sovereignty over the whole peninsula, factoring the cultural homogeneity of the peninsula into the equation, my view is that the Korean War was a civil war. That said, the nature of the war did change as the US and PRC came to shoulder the majority of the burden of actual fighting, but fundamentally I see it as a civil war which drew the attention and intervention of foreign powers who saw an opportunity to further their own goals in the area.