The Cook Islands had given up on its dream of United Nations membership and, with it, greater autonomy and influence on the world stage. New Zealand, the country that once governed the small Pacific nation and still exercises power there, had rejected its repeated entreaties.

But now that President Biden has provided a stamp of approval — with formal recognition for the Cook Islands at a summit of Pacific nations in September — the country of 18,000 people has shifted course, moving toward joining international bodies with renewed vigor.

What began with Washington’s attempt to counter Chinese influence with America’s imprimatur has now prompted an apparent shift in New Zealand policy and created a project of something resembling separate nationhood for the Cook Islands, even as its residents remain New Zealand citizens.

The Cook Islands’ prime minister, Mark Brown, said in an interview that the country now planned to apply for full membership in the International Monetary Fund “at least by next year” and that his diplomats were “talking to key, critical people” about U.N. membership.

“We put it on hold,” he said. “But the U.S.’ announcement,” he added, “has put this back on the agenda.”

Mr. Brown attributed Mr. Biden’s statement to an improved American understanding of the Cook Islands’ relationship with New Zealand. He also pointed to the fact that “the Pacific is a very contestable region right now” — a nod to the growing battle for influence between the United States and China.

Until recently, Mr. Brown said, Western countries neglected the Pacific, driving many countries to work with Beijing, even though “it’s no secret that China has a program of expanding its influence throughout the world through these development programs.”

Against that background, Mr. Brown wants I.M.F. and U.N. membership partly to ensure that his country can access new forms of support like a fund to help poorer nations cope with climate disasters, which richer nations agreed to create at a recent U.N. climate conference.

The Cook Islands, which lies halfway between Hawaii and New Zealand, has long had to navigate around larger powers. First annexed by Britain in 1901, it became a New Zealand colony soon after. In 1965, New Zealand allowed it to become self-governing in free association, a status short of full independence under which Cook Islanders hold New Zealand citizenship and New Zealand retains some responsibility for the country’s foreign affairs.

In 2001, Helen Clark, New Zealand’s prime minister at the time, announced that to apply for U.N. membership, Cook Islanders would have to give up New Zealand citizenship, a deal breaker for them. Ms. Clark’s successor, John Key, said in 2015 that New Zealand would not support U.N. membership for the islands.

“We just can’t see how the U.N. would ever support essentially people who hold New Zealand passports and therefore are not independent in the nature of independence having a separate seat,” Mr. Key said at the time.

This opposition was partly motivated by fears that U.N. membership “would diminish the basis on which New Zealand has the ‘right’ to exercise influence and seek control,” Caroline McDonald, a former New Zealand ambassador to Fiji, wrote in a 2018 doctoral thesis.

Ms. McDonald also wrote that New Zealand had previously marshaled opposition against U.N. membership for the Cook Islands by arguing that it could strengthen similar bids by Palestinians or the remnants of the colonial empires of the United States, Britain and France.

But after Mr. Biden’s announcement, New Zealand appears to have changed tack. Aupito William Sio, New Zealand’s minister for Pacific peoples, said that New Zealand “will back and support the aspirations” of its Pacific realm countries “as far as they ask of us.”

Nanaia Mahuta, New Zealand’s foreign minister, has denied any change in policy, but she also said that New Zealand had “signaled its willingness to engage” on U.N. membership for the Cook Islands.

Ms. Mahuta also indicated that New Zealand felt pressure following Mr. Biden’s statement, saying, “It has been clear since the U.S. formally recognized the Cook Islands that this has opened a door for deeper and more direct engagement within the international community.”

Mr. Brown, the prime minister, said he was confident that the United States’ recognition meant that any remaining barriers could be overcome. “It’s going to be very, very difficult,” he said, “to say that this country should not be a member of the U.N. or the I.M.F.”