No. Those were specific campaign names. The Desert War did not include Operation Torch. The Americans had no part in the battles of Tobruk nor El Alamein. Nor did they fight the Itaians and the Afrika Corps in Egypt and Libya. They didn't fight in the Quatara depression, nor did they do anything with the LRDG or the SAS. That was the Desert War, and it was entirely a Commonwealth campaign.
The American forces fought the Vichy French when they landed in Morocco and Algeria. It was only after the Vichy forces were defeated that the Americans faced the last of the Afirka Corps in Tunisia where the British and Commonwealth forces had forced them to concentrate by attacking from the East. With the British, American and Free French attacking from the West, and the British 8th Army attacking from the East,the Afrika Corps was finally forced to surrender during May of 1943.
The Americans were in Western North Africa from November of 1942 until May of 1943. The Desert War lasted from 1940 until October of 1942 with the second battle of El Alamein. The Americans weren't even there.