Asking when the Second World War started will net you a different answer depending on who you ask. There are some who argue that the war began in 1937, and some who argue for a later date, with the involvement on the United States being the usual event to turn a localised conflict into a World War.
1939 is the general consensus and there's a few reasons for this. If you ask any of my dear anthropologist friends, they will put it down to good old fashioned euro centrism, but I don't think this really holds water. It's important to remember that from 1939 to late 1941 there were two wars being fought at the same time, the second Sino-Japanese war that you mentioned and the war in Europe. This was not just a technical division, as there was no overlap in the belligerents at this stage. China was not at war with Germany and Italy, and Japan was not at war with Britain or France, or indeed the Soviet Union when Germany launched Barbarossa.
Because the wars are totally separate at this stage, saying the war started in 1937 leaves you with an unsatisfactory position of either declaring that the early fighting in Europe was not part of World War 2, or trying to combine two wars when at that stage they were really nothing to do with each other.
The solution of considering the invasion of Poland the start of the war seems the most acceptable to me. It was easier to describe the the European War as a World War at that point, mainly because it involved two global empires.After the invasion of the Soviet Union, the European front involved the major global economic powers of the time with the notable exception of the United States. This is not to belittle the war between Japan and China by any means, but it was a much more localised and focused war that gradually merged into the Second World War after Pearl Harbour and US involvement
The date of the start of WWII depends on your perspective. Here are the starting dates for the Big Four
America: 7 December 1941
Britain: 3 September 1939
China: 7 July 1937*
Russia: 22 June 1941
The Americans, Chinese, and Russians don't consider 1 September 1939 to be the start of WWII, although that was the day Germany invaded Poland. The war started with Pearl Harbor for America, with the Marco Polo Bridge for China, and with Operation Barbarossa for Russia. The British are the only ones who might consider the start date of WWII to be different from the start date of their involvement. But the war brought them together as the Allies and resulted in the same end date: 9 September 1945
Various other nations across the world would join at different times. For example, Mexico joined on 22 May 1942 and sent the Escuadrón 201 Aguilas Aztecas (Aztec Eagles), an aerial unit that flew against the Japanese.
*Some Chinese consider China's true start date to be 18 September 1931, when Japan invaded Manchuria