Congress holds the power to declare war, but in the wars with Korea, Vietnam, and both wars with Iraq, how have the presidents at the time been able to begin military action?

The President has used his role as commander-in-chief to initiate military actions, but not war.

Explanation:

The difference between a military action or intervention and war is diffuse and subject of debate among constitutional scholars. It seems that if the president sends military abroad without approval of Congress, then, it is a military intervention, and if the president does the same with approval of Congress (as it happened in World War II, the last time Congress declared war), then, it is war.

Therefore, the Korean, Vietnam, Iraq, and most recently, Libya and Syria interventions are not legally considered war, but military interventions.