The Sino-Japanese War did not in itself drag a big chunk of the nations of the world into a conflict - it was predominantly between China and Japan with some 'interested' onlookers (ie. aid went to the Chinese, but no one else really got directly involved).
The invasion of Poland on the other hand turned out to be a tipping point where a general European conflict followed, bringing in many more nations. In a way Japan's earlier commenced conflict got drawn into WWII, rather than being the initiator. Similarly Italy's invasion of Ethiopia 1935-1936 was an important precursor to WWII (being depicted as a blow to the status quo etc.), but wasn't really the 'beginning' of the broader conflict. Or there is the concept of WWI and WWII forming a single conflict - the 'European Civil War' or 'Long War'.
There are many elements to this question that make answering it difficult. Essentially it comes down to focusing on Germany as the epicenter of all the international conflicts which we categorize as "the Second World War." Claiming the war started in 1939 recognizes that the UK and France no longer saw appeasement as a viable solution to neutralize German aggression. Before Poland, Germany had invaded Czech Sudetenland, annexed Austria, occupied Denmark, etc. These were all dismissed under appeasement.
The Japanese invasion of China was somewhat disconnected from European politics. That's not to say that the UK was not concerned with the events in Asia, they most certainly were. Australia, India, Singapore, are all on the peripheral of Japan's Greater co-prosperity Sphere. their motivation for this sphere was to create an Asian sphere of influence, free of Western intervention. At least, that's what they tried to explain it as, but everyone knows its true ambitions were for empire. This conflict was considered, by Europeans as a side-show disconnected from the turmoil simultaneously happening in Europe which everyone was focused on. Prior to 1937, was the Spanish Civil War which began in '36. Italy had been very aggressive; brutally suppressing Libyan rebels, and even sending significant shipments of weapons/troops to join the Nationalists in the Spanish Civil War.
In my opinion, I would say the most accurate time to date the 'start' of ww2, would be at the Italian invasion of Ethiopia during the 2nd Abyssinian War (1935). As a result of this action, France and Great Britain imposed sanctions on Italy, who had attempted to expand her colonial African empire. Before these sanctions, Italy and Germany hadn't been particularly close, but afterwords, being forced to increase trade with Germany and sharing similar political ideologies greatly strengthened their relations. But that is me, assuming Italy as the epicenter of the conflicts culminated into "the Second World War."