However, as you said it is fuzzy: the hostilities were still somewhat limited even after the invasion in Manchuria; but total war broke out in July 1937 after the Marco Polo Bridge incident.
It directly contributed to sanctions such as American embargo of Japanese imports, and things like Export Control Act of 1940 and then the Tripartite Pact.
(BTW, thanks for asking; this made me read more about history in the area - I had no idea, for instance, that Soviet artillery and airplanes had attacked Chinese Muslim nationalist (KMT) troops with mustard gas in 1934 in the Soviet invasion of Xinjiang, nor that there were still White Russian troops that operated on same side with NKVD and GPU troops. What a murky business.)
Similarly, if you're counting political events that foment war as being part of that war, then WWII started at least as far back as 1919, when Ferdinand Foch stated that the WWI peace treaty "wasn't a peace, but an armstice for 20 years".
The thing is that Japan's military adventurism in China was a regional affair, conducted between neighbours, whereas Germany attacking Poland brought in countries from all inhabited continents except South America.
It's not so clear-cut where policy ends and war starts, and when a war turns into a global war, but I cannot say that September 1, 1939, would clearly be where we draw the line. The fighting in China in 1937 (Battle of Shanghai) was actually bigger in scope than the invasion of Poland in September 1939, and with more casualties. Then the Winter War in Finland exceeded these, with big losses for USSR, and then large-scale war started on the Western front in the Battle of France in 1940 which was even bigger than Winter War in terms of casualties.
nothing to do with muslims
Kuomintang had, among others, Muslim generals/warlords and troops, in this case I was referring to Ma Hushan's Hui (Muslim) troops.
https://en.wikipedia.org/wiki/Ma_Hushan
Compare to
https://en.wikipedia.org/wiki/Islamic_rebellion_in_Xinjiang_...
However, it bears some relevance to the matter that KMT-affiliated troops that fought against Soviet invasion in a predominantly Muslim-populated area were Muslims. I don't see here any particular parallels towards Islamic or Islamist movements today.
Edit: Canada already sent infantery units to Britain during the Phoney War (that was the period until the war in the west started:
"The Royal Air Force dropped propaganda leaflets on Germany and the first Canadian troops stepped ashore in Britain, while western Europe was under a period of uneasy calm for seven months.[8]" https://en.wikipedia.org/wiki/Phoney_War
To be picky, the Battle of the Atlantic started the same day as the British and French declaration of war, with the sinking of the SS Athenia. And the Norwegian Campaign began the month before the invasion of France.
As for the connection there was only one country that was fighting on two fronts at the same time: the US. Limiting the WW2 definition to when the US started to enter the War is reductive, I think.
"In effect, Australia fought two wars between 1939 and 1945[2] – one against Germany and Italy as part of the British Commonwealth's war effort and the other against Japan in alliance with the United States and Britain. While most Australian forces were withdrawn from the Mediterranean following the outbreak of war in the Pacific, they continued to take part in large numbers in the air offensive against Germany."
https://en.wikipedia.org/wiki/Military_history_of_Australia_...