But others contend that, words aside, Obama isn't following a radically new direction in combating terrorism.
"A lot of this is the rhetoric of change," said Gordon Johndroe, a spokesman for Bush's National Security Council. "At the end of the day, the policies are much more important [than the words]. And as long as the Obama administration continues to go after terrorists wherever they may be, then that's the most important thing."
The administration has hardly sought to advertise the changes in language. When an internal memo surfaced stating that the administration preferred the term "Overseas Contingency Operation" to "Global War on Terror," Pentagon spokesman Geoff Morrell said he "never received such a directive."
"Perhaps somebody within [the Office of Management and Budget] may have been a little over-exuberant," he said.
"I haven't gotten any directive about using it or not using it. It's just not being used," Clinton said during a briefing with reporters aboard her plane to the Hague to attend an international conference on Afghanistan.
"The administration has stopped using the phrase, and I think that speaks for itself," she said at a different point during her trip. "Obviously."
Some observers are interested to see what term replaces "war on terror."
"There's a reason we called it 'war on terror,'" Mark McKinnon, who served as a media adviser to Bush, told ABCNews.com. "I think people notice a difference, not so much the specifics. But ... they sense the optics of difference, which is particularly important, especially since many of the policies, like Afghanistan, are not significantly different than [Bush's]."
Michael O'Hanlon, a senior foreign policy fellow with the Brookings Institution, said the term "war on terror" had lost much of its utility even before Obama took office. Now, he said, the catchall phrase is less useful because the public is accustomed to referring to individual operations, such as the wars in Iraq and Afghanistan, or the efforts to help Pakistan root out terrorists.
He said it makes sense for Obama to drop Bush-era language, given how unpopular Bush's policies were in the world. But he added that eliminating phrases or changing them won't matter in the long run.
"It makes sense for a new president to try to change some of the semantics. You might as well take the low-hanging fruit where you can reach it," O'Hanlon said. "But either you need a better term, or you need for the whole concept to be less important and relevant, in which case you can't take too much credit for [changing the terms]."
Words may matter, but as always, it will be the president's the actions that count.
Said Johndroe, the former NSC spokesman: "In fact, it's everybody's war. It's not Obama's war, it's not America's war, it's the entire civilized world against extremists. And that's why if the rhetoric change helps bring everyone along to counter extremism, then that's OK."