GENEVA — President Biden had three big tasks to accomplish on his first foreign trip since taking office: Convince the allies that America was back, and for good; gather them in common cause against the rising threat of China; and establish some red lines for President Vladimir V. Putin of Russia, whom he called his “worthy adversary.”

He largely accomplished the first, though many European leaders still wonder whether his presidency may yet be just an intermezzo, sandwiched between the Trump era and the election of another America First leader uninterested in the 72-year-old Atlantic alliance.

He made inroads on the second, at least in parts of Europe, where there has been enormous reluctance to think first of China as a threat — economically, technologically and militarily — and second as an economic partner.

Mr. Biden expressed cautious optimism about finding ways to reach a polite accommodation with Mr. Putin. But it is far from clear that any of the modest initiatives the two men described on Wednesday, after a stiff, three-hour summit meeting on the edge of Lake Geneva, will fundamentally change a bad dynamic.

Mr. Biden, one of his senior aides said after the meeting was over, “is perpetually optimistic” that Mr. Putin may, despite a long history of efforts to undermine the Western alliance, see advantage in changing course.

“He may be the only one,” the aide said.

This was Mr. Biden’s European comeback tour, and he began in England, on the rocky shores of Cornwall, playing all the old crowd favorites — talking about friendship, alliances, consultation, comity and multilateralism. At every stop he opened with the same three words: “America is back.”

He quoted poets, mostly Irish poets. It was all warmly received by European leaders, who had been battered and bruised by President Donald J. Trump’s attacks on them for being weak, divided, self-interested free-riders.

What Mr. Biden did not say was almost as important as what he did say. He did not ask why he should commit to defending countries that run trade surpluses with the United States, a frequent theme for Mr. Trump. Instead, he spoke of the economic benefits of developing new forms of clean energy or joint projects in semiconductor manufacturing.

Yet, when President Emmanuel Macron of France said as he sat with Mr. Biden that “it’s great to have the U.S. president part of the club,” it was a line that clearly would play differently in different parts of a divided United States. Among the 74 million who voted for Mr. Trump last year, the “club” is the problem, a place where American interests get subjugated.

But Mr. Biden never directly addressed — at least in his public remarks — the fundamental source of Europe’s post-Trump traumatic stress syndrome: doubts about the future of American democracy. Obviously he cannot offer any predictions, much less guarantees, about what will happen when his term runs out in January 2025. So he didn’t try.

“Don’t underestimate the Trump years as a shock to the E.U.,” said Rosa Balfour, the director of Carnegie Europe, a Brussels think tank. “There is the shadow of his return and the E.U. will be left in the cold again. So the E.U. is more cautious in embracing U.S. demands.”

But Mr. Biden has argued to the Europeans that the best insurance against another Trump-like president is to work with him to show that democracies work, and to respond to the China challenge.

The competition with China lay at the center of a deal to resolve the decade-long Boeing-Airbus dispute, a source of tariffs and recriminations that dates back to 2004.

What finally resolved it — and wiped away implementation of $11.5 billion in tariffs — was a common resolve to avoid dependence on a Chinese supply chain for building planes and to slow China’s entry into the commercial aircraft business. The subtext was to begin to engage Europe in “decoupling” from China’s economic influence.

While there is a palpable sense of relief at the message that America is back, said Thomas Bagger, a German diplomat who is an adviser to the country’s president, “we also have noticed that the center of gravity of U.S. policy is changing, and the centrality of the rise of China to U.S. interests will have profound consequences for Europe and any new German government.”

Both Mr. Macron and Chancellor Angela Merkel of Germany — who has always been the strongest voice for treating China as a partner first and a competitor second — expressed some concern that a balance be struck on China, which is an important trade partner, crucial to solving the climate crisis and not a military power in Europe.

“If you look at the cyberthreats and the hybrid threats, if you look at the cooperation between Russia and China, you cannot simply ignore China,” Ms. Merkel said. But she also said: “One must not overrate it, either — we need to find the right balance.”

Another subtext of the trip was the discomfort of some European leaders with Mr. Biden’s repeated declarations that the struggle of the age is “democracy versus autocracy.” It is not that they disagree, several said on the sidelines of the meetings, but rather that Mr. Biden’s words could harden the division and usher in a new Cold War.

They say they understand Mr. Biden’s concern that China’s technology strategy is all about building a system of cellular networks, undersea cables and space assets that would give it the capability to cut off or secretly monitor communications.

And they do not argue with the White House effort to halt American investment in Chinese firms that are selling the facial recognition software and social-scoring algorithms that Beijing uses to repress dissent and imprison its Muslim minority. But so far they have not joined Secretary of State Antony J. Blinken when he refers to Beijing’s actions against the Uyghur population and other predominantly Muslim ethnic minorities as genocide.

So Mr. Biden toned down his autocracy vs. democracy talk for this trip. And that worked.

Yet while “Biden has gotten words from the Europeans, he hasn’t gotten deeds,” said James M. Lindsay, director of studies at the Council on Foreign Relations. “Settling some trade issues is a very good start. But it’s not how you start, but how you finish, how you translate the sentiments in the communiqués into common policies, and that will be very difficult.’’

Mr. Biden carefully choreographed the trip so that he demonstrated the repairs being made to the alliance before going on to meet Mr. Putin. Mr. Biden made clear he wanted to present a unified front to the Russian leader, to demonstrate that in the post-Trump era, the United States and the NATO allies were one.

That allowed Mr. Biden to take a softer tone when he got to Geneva for the summit meeting, where he sought to portray Mr. Putin as an isolated leader who has to worry about his country’s future. When Mr. Biden said in response to a reporter’s question that “I don’t think he’s looking for a Cold War with the United States,’’ it was a signal that Mr. Biden believes he has leverage that the rest of the world has underappreciated.

Mr. Putin’s economy is “struggling,’’ he said, and he faces a long border with China at a moment when Beijing is “hellbent” on domination.

“He still, I believe, is concerned about being ‘encircled,’ ” Mr. Biden said. “He still is concerned that we, in fact, are looking to take him down.” But, he added, he didn’t think those security fears “are the driving force as to the kind of relationship he’s looking for with the United States.”

He set as the first test of Mr. Putin’s willingness to deal with him seriously a review of how to improve “strategic stability,’’ which he described as controlling the introduction of “new and dangerous and sophisticated weapons that are coming on the scene now that reduce the times of response, that raise the prospects of accidental war.”

It is territory that has been neglected, and if Mr. Biden is successful he may save hundreds of billions of dollars that would otherwise be spent on hypersonic and space weapons, as well as the development of new nuclear delivery systems.

But none of that is likely to deter Mr. Putin in the world of cyberweapons, which are dirt cheap and give him an instrument of power each and every day. Mr. Biden warned during his news conference that “we have significant cyber capability,” and said that while Mr. Putin “doesn’t know exactly what it is,” if the Russians “violate these basic norms, we will respond with cyber.”

The U.S. has had those capabilities for years but has hesitated to use them, for fear that a cyberconflict with Russia might escalate into something much bigger.

But Mr. Biden thinks Mr. Putin is too invested in self-preservation to let it come to that. In the end, he said, just before boarding Air Force One for the flight home, “You have to figure out what the other guy’s self-interest is. Their self-interest. I don’t trust anybody.”

David E. Sanger reported from Geneva and Steven Erlanger from Brussels.