How were Korea and Vietnam "Proxy Wars"? Why was it a proxy war and not another kind of war?
I would say because two superpowers used other nations to fight without fighting face to face.
In Korea, the US army together with the South Korean army (and some other nations under UN coalition) fought indirectly against the Soviet Union engaging the North Korean and Chinese armies.
Again, in Vietnam, the US army fought indirectly against the Soviet Union this time supporting the South against North Vietnam.
In both cases you can see that no American killed openly in the battle field any Soviet soldier and vice-versa and the two nations never declared war to each other.
In a "normal" war American and Russian would have faced each other in the battlefield and probably ended up using nuclear weapons in the process!
So, although cynically horrible, these proxy wars probably avoided a nuclear holocaust of planetary scale.