Asked now that he's met face to face with Putin if he thinks can trust him, Biden said the summit was "not about trust."

"This is about self-interest and verification of self-interest," Biden said. "Almost anyone that I would work out an agreement with that affected the American people's interest, I don't say, 'Well, I trust you, no problem. Let's see what happens.' You know, as that old expression goes, 'the proof of the pudding is in the eating.'"

Biden said he raised with Putin -- and will continue to raise -- cases like jailed opposition leader Alexei Navalny, the two "wrongfully" imprisoned Marine veterans, Paul Whlean and Trevor Reed, and the ability of Radio Free Europe and Radio Liberty to operate, for starters.

However, he did not claim he caused Putin to change his behavior on those matters, signaling he thought the face-to-face meeting in itself a success.

"There were no threats, just simple assertions made. And no, 'Well, if you do that then we'll do this' with anything I said. It was just letting him know where I stood, what I thought we could accomplish together, and what, in fact, if there were violations of American sovereignty, what would we do," Biden said of the meeting.

On cyberattacks, Biden said they agreed to task expert in both countries "to work on specific understandings about what's off-limits and to follow-up on specific cases that originate in other countries." Putin refuses to say Russian hackers are to blame for some of the recent cyberattacks, despite U.S. intelligence indicating otherwise.

On the Middle East, Biden said Putin raised the withdrawal of U.S. troops in Afghanistan. He also said they agreed it's in the interests of both nations not to let Iran acquire nuclear weapons.

Biden declared the summit a success at his solo press conference, saying "I did what I came to do."

"Number one: identify areas of practical work our two countries can do to advance our mutual interests and also benefit the world. Two: communicate directly, directly, that the United States will respond to actions that impair our vital interests or those of our allies. And three: to clearly lay out our country's priorities and our values so he heard it straight from me," he said.

"There's much more work ahead. I'm not suggesting that any of this is done," Biden added, before taking questions. "But we've gotten a lot of business done on this trip."

Following Putin's preser, Biden delivered his readout on the meeting in a solo news conference from outside his hotel in Geneva.

He started with a joke about that chaotic photo op earlier, in which Russian security pushed American reporters, before reading from prepared remarks giving his take on the summit following at the end of his week abroad.

"I've just finished the -- the last meeting of this week's long trip, the U.S.-Russian summit. And I know there were a lot of hype around this meeting, but it's pretty straightforward to me," Biden said, that "there's no substitute ... for face-to-face dialogue between leaders, none."

"President Putin and I -- share a unique responsibility to manage the relationship between two powerful and proud countries, a relationship that has to be stable and predictable," he continued. "We should be able to cooperate where it's in our mutual interest. And where we have differences, I wanted President Putin to understand why I say what I say, and why I do what I do, and how we'll respond to specific kinds of actions that harm America's interest."

Biden said he told Putin that his agenda isn't "against Russia or anyone else" but "for the American people."

When Putin meets Biden on Wednesday in Switzerland, experts in Moscow say for all their differences, the two leaders want something similar from their first summit: to cool things down.

The U.S. and Russia's relations are the worst they have been since the Cold War and since 2016 in particular seem locked in almost permanent crises.

Biden has said he wants a more stable and predictable relationship with Russia, one that would allow it to focus on other foreign policy priorities that are more important to it, like taking a harder line with China. The Kremlin for its part has faced a continuous and intensifying barrage of sanctions-- the latest in April-- and with its crackdown on opposition at home and aggressive actions abroad is increasingly becoming a pariah with western countries.

Since coming to office, Russia has appeared to want to get Biden's attention. The president offered Putin the summit after Russia massed thousands of troops on Ukraine's border in April.

But now, having got Biden to the table, analysts said Putin has a clear proposal to deliver in Geneva: stay out of Russian domestic politics and Russia might act less troublesome abroad.

"The Kremlin wants to transition to a respectful adversarial relationship from a disrespectful one we have today," said Vladimir Frolov, a former diplomat at Russia's embassy in Washington and now a commentator on foreign affairs.

"That is, it wants to be treated the same way the Soviet Politburo was treated by the US in 1970-80s," Frolov told ABC News. "Meaning no name-calling" — such as Biden calling Putin a "killer" — "no personal sanctions on the leadership, no democracy lectures, regular personal summit meetings; respectful tone of discussions, no tangible support for Russian opposition."

It will not be an invitation for détente but instead to return to the later years of the Cold War when Putin was a KGB agent and the Soviet Union and the U.S. saw each other as enemies but tried to maintain a predictable relationship. And, crucially, where Russia was treated as an equal.

"For this, the Kremlin is prepared to promise to behave more responsibly," Frolov said.

"This seems to be in line with what the White House sees as a desirable deliverable," he continued. "So unless one of the leaders stormed out of the meeting shouting expletives, the summit would be a major success."

-ABC News' Patrick Reevell