Public relations (PR) has become an industry devoted to obtaining (or avoiding) free publicity for its clientele. The simplest and most obvious publicity stunt is the press release. If you want to get attention it is the least expensive way of attempting to get the media interested. The next most common approach is to arrange a press conference in the hope that the topic will attract the press and the outcome will be favourable reports on the event. Specialised events arranged by PR people range from book or new car launches to award ceremonies. The Academy Awards are a gigantic publicity stunt to promote the movie industry, its people and products. In the same vein, showbusiness types staging weddings with invited press do so to promote their careers.
Amateur stunts can be trivial or deadly serious. Among the trivial are students occupying a university building to highlight grievances, politicians progressively releasing leaked material to boost their profiles, software companies challenging hackers, a radio station dropping live turkeys from a helicopter on Thanksgiving, etc. Serious publicity stunts include terrorist attacks, strikes, mass demonstrations, hijacks, kidnappings, hunger strikes, suicides, and murders.
Many people would maintain that attempts to raise awareness of serious causes are not "stunts". That depends on the point of view of the observer. The toppling of Saddam Hussein statues in front of the media in Iraq was a publicity stunt but the motives behind the show were serious to those who planned the events.
One way of perhaps drawing a distinction might be to label as stunts, events specifically designed to attract publicity. Events designed to gain an objective and which incidentally attract publicity can be exempted from the term. For instance, if an animal rights enthusiast were to rescue a dancing bear in India and that action became known through a report on a court case, that would not be a publicity stunt. If however, the activist arranged for the press to cover the rescue, it would be a publicity stunt.