Why is Winter Solstice (shortest day) considered the beginning of winter?

Two-thirds of the earth is covered by oceans. Water being transparent, can store heat energy quite deep. Likewise, when it gets colder, water that is about 300 times denser than air, releases that energy. The oceans are, actually, the great heat regulators on earth.

Because of that, the coldest part of the year is not at, but after the shortest day. But it varies from what we call, maritimes climate (near a shore and later delay effect) to continental (inland and sooner cold).

You will probably get answers telling you that the winter doesn't start at the winter solstice (last night, at 23:05 UTC) but the first of January. Well, all the meteorological books and the nautical pilot books, with their climatic tables, I have seen in the past 50 years (yes, I am an old man! ;-) use the solstices and the equinoxes as the definitions of the seasons. But it is possible that different authorities have different standards. What is certainly not scientific is my wife's definition of the first day of the winter. She means that it is the day of ... the first snowfall. ;-)

On that day, the amount of daylight is the shortest of any day during the year. But it will increase each day now until the first day of summer which will be June 21, 2015.

Winter is a time period not the weather. the weather tends to follow the time periods somewhat but does not define the time periods. the seasons are divided the way that they are because of observable changes in the position of the sun. furthest south, furthest north, and at the equator (twice per year).

Because,only on that day,the amount of solar radiation received in the Northern hemisphere is the least.