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."

Amid all the high-level shadow boxing setting up President Joe Biden's summit with Russian President Vladimir Putin, Biden has added a new wrinkle -- one that amounts to a test for himself that awaits him back home.

Biden has cast this moment in the world community in broad terms for the United States -- a chance to assert the power of democratic nations in the face of challenges from China and Russia in particular. Asked Monday what he is telling allies who may be worried about any American slide toward autocracy, Biden again went big.

"What I'm saying to them is, watch me," Biden said. "That's why it's so important that I succeed in my agenda."

Biden was nonchalant in his condemnation of what he called the "phony populism" of former President Donald Trump. Speaking about Republicans, he flatly observed that "the Trump wing of the party is the bulk of the party, but it makes up a significant minority of the American people."

Still, just hours earlier, Senate Minority Leader Mitch McConnell served notice that his brand of hardball is still going to be played, with a warning about what Republican Senate control would mean for any Supreme Court vacancy under a Democratic presidency.

McConnell is objecting to congressional scrutiny of Trump-era Justice Department strategies, just like he did to the proposed Jan. 6 commission. And it's still far from clear whether any infrastructure or climate-change legislation can pass with Republican support, to say nothing of the prospects for tax reform.

Asked about Putin's laughing response to Biden's assertion that he is a killer, Biden said his message back would be that he is laughing as well. The world now is watching -- and will still be when Biden and Putin are both back home.

-ABC News Political Director Rick Klein