This is a subjective definition, one I've heard used in different contexts in numerous places, so I understand if there isn't a straight answer here.
Usually when I think of a start date, I go to three years:
1945: End of World War II, beginning of Cold War, end of colonialism, beginning of the USA's emergence as a superpower.
1991: Fall of the USSR, modern geopolitical climate as the US' sole superpower begins to take shape.
2001: September 11 attacks; U.S. War on Terrorism and surveillance begins in earnest.
Are any of these three dates accurate? Is there another date you would use?
It depends. When talking about international politics, 1648 (the Peace of Westphalia) is often mentioned because it meant the beginning of an international system of states, rather than feudal dynasties rulling swathes of land.
Another date would be 'the 19th Century'. It's a rather ambiguous date, certainly not 'exact', but the events of the 19th century shaped the modern world. The independence of the South American states, the unification of Italy and Germany, the emergence of parliamentary democracy/monarchy, the various Isms, the Imperialism that helped shape the modern day, and so on.
What you are aiming at, however, is "when does contemporary history begin"? Some rigid and, I think, bullshit definition is the informal Twenty Year Rule. Others would argue everything that is in living memory is contemporary, so... With the oldest person alive being 116, the contemporary period would start in 1898.
It eventually depends on your personal definition of "contemporary" or "modern", really.
Edit
As for the second question. I would accept 1945 as a valid contestant for a temporal discontinuity. Our current world is still heavily influenced by the events just prior and just after 1945. 1991 and 2001 - less so. Personally, 1991 could maybe qualify, but 2001 certainly not.
Academic history is broadly split into three periods: medieval, early-modern and modern. The dates which these periods cover can vary but broadly speaking the medieval would cover 476AD-1500, the early-modern covers 1500-1800, and modern covers 1800-present.
The label 'contemporary history' is sometimes used too which would cover events post-1945.
The French Revolution, which occurred in 1789, and the ideas it embodied are sometimes seen as a pivotal date in the shift towards modernity. The Industrial Revolution, which began in around 1760, is seen as important in the move towards modernity too.
As stated by the others, historical time period dating gives 1815 as the beginning of the Modern Age. The reason was that with the fall of Imperial France and the International anger that France backed Napoleon during the 100 Days, The Congress of Vienna limited France to her 1789 borders (compared to the 1796 borders), so it marked the loss of French hegemony and the rise of Prussia.
The rise of Prussia is a dominant topic because Prussia and Germany lead to the World Wars. The end of the Second World War is the beginning of the Post-Modern Age because it is the beginning of another series of contests (US vs. USSR).
2001 could be a separate new age but that was thirteen years ago and that's well within our 20 year rule and the history is no where close to being fully written.