CINCINNATI — Hillary Clinton offered a vigorous case on Wednesday for America’s place as a singular and vital leader in the world, outlining an approach to global affairs that differs sharply from the “America First” posture espoused by Donald J. Trump.

“If there’s one core belief that has guided and inspired me every step of the way, it is this: The United States is an exceptional nation,” Mrs. Clinton, the Democratic presidential nominee, said in an address to the American Legion’s national convention here.

In her speech, Mrs. Clinton championed the notion of American exceptionalism, a term traditionally embraced by Republicans. And, as she warned about the dangers of electing Mr. Trump, she offered herself as a well-prepared alternative — for Republicans as much as Democrats.

While the campaign of Mr. Trump has been marked by a hard-line nationalist approach, Mrs. Clinton sketched a view of America as a model for other nations and a force for good in the world, saying that such a role is a “serious responsibility.”