@Finn dAbuzz,
Gracie's teacher is correct. The cold war only involved shooting wars which were proxy wars. The United States and the United Nations fought the North Koreans and the Chinese, supported by the Soviet Union, but they did not fight the Soviets. The United States, the Republic of Korea and Australia fought the Vietnamese, who were supported by the Soviet Union, but they did not fight the Soviet Union. The Soviets fought the Afghan
mujahadeen, who were supported by the United States, but they did not fight the United States.
In the Second World War, by contrast, quite apart from the European nations who fought one another, directly, the Chinese fought the Japanese, with American support, while the United States directly fought the Japanese. Thailand was an ally (probably very reluctantly) of Japan, but they caved in before being invaded. In the Americas, Canada (as a part of the British Commonwealth), the United States and Brazil fought the Germans. Canadian troops briefly fought the Japanese at Hong Kong, but were forced to surrender and were interned. Otherwise, Canadian troops fought in Italy, France and Holland, and the Royal Canadian Navy was a crucial part of the Battle of the Atlantic. The RCN escorted more convoys across the Atlantic than the Royal Navy and the United States Navy combined. By the end of the war, with the destuction of the Japanese Imperial Navy, the RCN was the third largest navy in the world.
When, a few days after the attack on Hawaii, Germany and Italy declared war on the United States, the United States directly fought German and Italy. The Germans had had supply ships interned in Spain at the beginning of the war, but the fascists in Spain didn't try to hard to enforce the internment, and those ships slipped away. (Spanish "volunteers" fought with the Nazis in Russia.) Some of them resupplied German submarines in the mouth of the Amazon river, and German submarines attacked Brazilian shipping. The government of the Brazilian dictator (you can look him up, Gracie, i don't recall his name) was reluctant to enter the war, but the popular feeling was with the Allies and against Germany and Italy, so the government finally declared war on the Axis in late 1942. The Brazilians sent a huge, reinforced infantry division to Italy to fight with the United Nations command, which division was attached to Mark Clark's Fifth U. S. Army. They also sent two fighter groups, flying American aircraft supplied by the United States.
So, nations from all over the world (including Africa--the South Africans--all whites--sent an armored division which fought in North Africa, and then in Italy) fought the Axis powers directly. These were not proxy wars, it was world-wide warfare, which began in China in 1937.
Gracie, your teacher is right, and was not at all being a pedant. She was being sensible, and demonstrated that she knows the history and understands its significance, and the relevant distinctions.