Im shocked you dont think the US is a shining example. Pray tell what nation you think IS a shining example?
The United States does not have to be a shining example of "best" in order for it to be exceptional. Exceptional is, as per the Wiki that you cite heavily (which may have been derived from another text I will recommend later on), based on the notion that the United States or the land prior to unification circa the 1770s was
different. Being different is exceptional, whether for better or for worse. Point in fact, if the United States has a higher per capita rate of gun violence than most or, say, all "developed nations" or whatever other standard, that is a demonstration of American Exceptionalism. If Americans continue to believe that their lot in life is based more on individual initiative and hard work than other "developed nations" or "Europe," that too is an affirmation of American Exceptionalism. If individuals in the United States have a statistically higher chance of seeing themselves as members or aligned to their nation-state than the rest of the "developed nations" or "European countries," this is also an affirmation of American Exceptionalism.
In terms of whether or not the United States has a divine mission, or perhaps some derivative Hegelian destiny to propel the world forward in History (with the capital H), that is
also American Exceptionalism.
Furthermore, the widespread
belief of American Exceptionalism gives credence to American Exceptionalism.
There were two great, shall we say, neoconservative entries on the modern conception of American Exceptionalism. Seymour Martin Lipset,
American Exceptionalism: A Double Edged Sword, and Peter H. Shuck and James Q. Wilson,
Understanding America: The Anatomy of an Exceptional Nation.
Another great one to read, since it's available everywhere, is John Winthrop's speech, "A Model of Christian Charity." Pay particular attention to the somewhat martyrdom he puts their experiment with. Americans have long held the belief that if they fail to live up to their high standards (standards which change over time), not only will they fail, but the world will fail,
and God (or History) will make an example of us falling from grace. We put a lot of pressure on ourselves to do right. Hence, we Americans can't merely act like everyone else...we have to be "better" than the rest. This means when liberals reacted negatively to "enhanced interrogation," they too were relying on American Exceptionalist rhetoric. To them, it may be true that barbaric countries (or perhaps the rest of the world, if they were given the opportunity) use torture, but "we Americans" absolutely cannot, because we are "above that."