Are there exact dates of "modern history"?

That depends on how exact you want to get. Modern History is taken to mean history from the Middle Ages forward. As such, early modern history began in the 1500s. Exactly what day of what year might be open to debate, but over the long haul of human history and prehistory, that’s a pretty definitive time period.

I think there are not clearly definable dates for “modern history”; it would vary from historian to historian and country to country. However, in this case, I suspect that what they really mean is since weather records have been kept. Looking quickly at a few wikipedia articles, it seems that weather recording began in the 1880s, so I would conclude that that is what the news story really meant.

I concur with @janbb.

It’s diffucult to tell what they mean. As @janbb said, formal records of weather began in the mid to late 19th century. Some eastern US cities have anecdotal (newspaper story) weather histories going back 250+ years, but not as day-by-day formal records of a fixed set of phenomena.
In a general sense “late modern history” is a moving target, usually considered the last hundred years. It’s a problem I’m encountering now in classifying my thesis, as it spans the time period roughly 1890–1914. I suppose that I’ll just go with whatever my thesis advisor recommends.

April 7, 1954 – October 23, 1986.