No one came seem to agree. Does the rainy season start in June or July? Does it end in July or run through August.
Online sources say one thing but talk with Koreans and they'll tell you another. If the past several very humid days with rainy evenings and nights have been any indication, then the rainy season is well underway. Check out the weather around Korea here.
Or just go straight the source, The Underground for five day forecasts. Thursday has a 100% chance of precipitation in the Inchon area. Can anyone really be that confident to guarentee 100% chance? Either way, I'll be sure to pack my umbrella.