As exceptionalism, this is a bunch of bull manure. This is being special in the everyone gets a trophy kind of special. If you're a nation, you're exceptional in this fashion. Everyone's got a culture they think made them great, there's nothing in American culture that doesn't have an analog elsewhere. While these cultural traits can help define why we're not exactly like any other individual nation they don't explain any more for us than why France is different from Germany is different from China is different from South Africa. This sort of cultural explanation explains why we get a UN membership card and aren't simply an administrative district; it tells us nothing about why we're great among nations. It also doesn't tell us why we can or can't adopt a specific policy. Tie any outcome back to American culture and you'll have an analogous situation elsewhere, there's nothing exceptional about particular policies.
Which isn't to say that American exceptionalism doesn't help explain why we're great among nations. It's just an American exceptionalism that is hard to swallow for those wishing for a cultural explanation or nativism.
In my reading a few months ago I came across a good section on American exceptionalism, a way of defining it that I think does represent our actual history and explain our greatness, which the normal use of the term exceptionalism does not, on either count. The section below is from John Gerard Ruggie's Constructing the World Polity, "Interests, Identity, and American Foreign Policy." 1998. p. 218.
America's form of nationalism differs from that of most other nations however. Most nations claim an "organic" basis in either land or people, and these are the usual referents of a nation's foundational myths. The American form of nationalism, in contrast, has no such organic basis.
...America traditionally has viewed itself as a willful community, or an elective community. ... In principle, anyone can become an American. But that fact is made possible, in turn, only because the American concept of political community rests on, not the exclusive organic specificities of traditional nations, but, in the words of political theorist Tracy Strong (1980: 50), "a universal or general foundation open in principle to everyone."
American nationalism, then, is a civic nationalism embodying a set of inclusive core values: intrinsic individual as opposed to group rights, equality of opportunity for all, anti-statism, the rule of law, and a revolutionary legacy which holds that human betterment can be achieved by means of deliberate human actions, especially when they are pursued in accordance with these foundational values. Being an American is defined as believing and doing these things.
This definition of American nationalism I believe accurately represents America's history and culture. I also think it clearly shows how our exceptionalism contributed to making us exceptional among all nations (though Ruggie is careful to state in a note that he sees this as distinctiveness, not exceptionalism; I'm more of a nationalist than he is). This form of exceptionalism really does bring out how America is unique not in the sense that every national culture is unique but how we truly are unique among nations.
This is why some of the cultural forces waxing strong worry me. The wish to precisely define who an American is and what being American means and to tie American identity more specifically to a codified set of myths and beliefs threatens the very heart of what makes us different from every other nation by making it clearer why individual Americans are distinct from individuals from any other nation.
Defining being American in cultural terms replaces American exceptionalism with the kind of organic myth of nation that every other nation has. The exceptional thing about America is that we are a community based upon a set of enlightenment ideas rather than pre-modern traditions and myths, open to any individual that believes in them and will act upon them, rather than a people defined by shared myths, background, and territory. We are defined by our inclusivity, not our exclusivity. By who we are and the beliefs we share rather than by who we aren't and the beliefs we disagree on. We are bound together by our shared beliefs in the founding ideals of our Republic, not in a national myth about our background or in strict adherence to a set of legal strictures. Another exceptional aspect is that these ideas are so firmly grounded in the ideas behind the enlightenment, we don't represent ancient traditions such as religion or heredity but instead are a nation firmly grounded in the ideas that gave rise to modernity and that are open to all, whatever their traditions. We may not see eye to eye in how to interpret and act upon these ideas in our modern world, or even how these ideas acted in our own history, but our shared identity is rooted in our mutual recognition that we are seeking to put these ideas into practice in the world as we find it.
None of this is to say that this modern trend to redefine being American in cultural terms is unique. It isn't, the desire to redefine America in terms that draw from the same roots that other nations' nationalism draw from is a tendency that is visible throughout American history. People throughout our history are made uncomfortable by our exceptionalism and desire to sacrifice our differences to be more like other nations and to enjoy the same kind of exclusionary certainties that are the privileges of those with more typical national myths. I have faith in the resilience of our beliefs and do not think this will occur but I do think there is a significant enough threat that it is necessary for us to speak out against the possibility of us adopting a typical national identity, like that which the Tea Party is seeking to promote. Go down this road and American exceptionalism is gone forever.