WASHINGTON — If President Trump emerged from his meeting with President Vladimir V. Putin of Russia last week hoping he had begun to “move forward” from the controversy over the Kremlin’s election meddling, as advisers put it, his flight home the next day made clear just how overly optimistic that was.

As Air Force One jetted back from Europe on Saturday, a small cadre of Mr. Trump’s advisers huddled in a cabin helping to craft a statement for the president’s eldest son, Donald Trump Jr., to give to The New York Times explaining why he met last summer with a lawyer connected to the Russian government. Participants on the plane and back in the United States debated about how transparent to be in the statement, according to people familiar with the discussions.

Ultimately, the people said, the president signed off on a statement from Donald Trump Jr. for The Times that was so incomplete that it required day after day of follow-up statements, each more revealing than the last. It culminated on Tuesday with a release of emails making clear that Mr. Trump’s son believed the Russian lawyer was seeking to meet with him to provide incriminating information about Hillary Clinton as “part of Russia and its government’s support for Mr. Trump.”

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The Russia story has become the brier patch from which the president seemingly cannot escape. It dominated his trip to Europe last week and, after he leaves on Wednesday night for a couple of days in France, it may dominate that trip as well. Every time Mr. Trump tries to put the furor behind him, more disclosures thrust it back onto the Washington agenda.

Even before the latest reports, Mr. Trump’s head-spinning willingness on creating a joint cybersecurity team with Russia fueled criticism. Now people close to the president and to his legal effort are engaged in a circular firing squad, anonymously blaming one another for the decisions of the last few days.

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The emails, which the younger Mr. Trump released after learning that The Times had obtained copies and was about to publish them, undercut the president’s line of defense in the Russia inquiry. For months, Mr. Trump has dismissed suspicions of collusion between Russia and his team as “fake news” and a “total hoax.” His eldest son, likewise, had previously asserted that talk of collusion was “disgusting” and “so phony.” Donald Trump Jr. said in a Fox News interview that he would have done things differently in retrospect, but he maintained he had done nothing improper.

At a minimum, however, the emails show that the younger Mr. Trump was not only willing, but also eager, to accept help advertised as coming from the Russian government. “I love it,” he wrote.

Joining him at the meeting with the Russian lawyer in June 2016 were Jared Kushner, his sister’s husband and now a senior White House adviser, and Paul J. Manafort, then the campaign chairman and a veteran political operative with longstanding ties to a pro-Russia party in Ukraine. Both the younger Mr. Trump and the Russian lawyer, Natalia Veselnitskaya, have said in recent days that no incriminating information about Mrs. Clinton was actually passed along during the meeting at Trump Tower, but that Ms. Veselnitskaya did discuss American sanctions imposed on Russian human rights offenders.

While Donald Trump Jr. has been the main focus of the controversy because he set up the meeting, Mr. Kushner faces potential trouble because he currently works in the White House and neglected to mention the encounter on forms he filled out for a background check to obtain a security clearance. The emails were discovered in recent weeks by Mr. Kushner’s legal team as it reviewed documents, and they amended his clearance forms to disclose it, according to people briefed on the developments, who like others declined to be identified because of the sensitive political and legal issues involved.

Similarly, Mr. Manafort recently mentioned the meeting to congressional investigators looking into possible collusion, according to the people briefed on the matter.

The disclosure of the emails left the White House again on the defensive as tension inside the president’s orbit has grown. Mr. Trump is exasperated and, at the urging of advisers, he said nothing publicly in defense of his son until Tuesday, when he issued a one-sentence statement. “My son is a high-quality person and I applaud his transparency,” Mr. Trump said. Hours later, he added on Twitter that his son was “a great person who loves our country!”

Advisers said the president was annoyed not so much by his son as by the headlines. But three people close to the legal team said he had also trained his ire on Marc E. Kasowitz, his longtime lawyer, who is leading the team of private lawyers representing him. Mr. Trump, who often vents about advisers in times of trouble, has grown disillusioned by Mr. Kasowitz’s strategy, the people said.

The strain, though, exists on both sides. Mr. Kasowitz and his colleagues have been deeply frustrated by the president. And they have complained that Mr. Kushner has been whispering in the president’s ear about the Russia investigations and stories while keeping the lawyers out of the loop, according to another person familiar with the legal team. The president’s lawyers view Mr. Kushner as an obstacle and freelancer more concerned about protecting himself than his father-in-law, the person said. While no ultimatum has been delivered, the lawyers have told colleagues that they cannot keep operating that way, raising the prospect that Mr. Kasowitz may resign.

Also, the president has fumed to close allies that he is mulling a staff change, and some members of his family have zeroed in on the chief of staff, Reince Priebus. But most Trump advisers privately concede that major changes are unlikely anytime soon.

The developments provoked sharp criticism by Democrats and even some Republicans. “Nothing’s proven yet, but we’re now beyond obstruction of justice in terms of what’s being investigated,” Senator Tim Kaine, Democrat of Virginia and Mrs. Clinton’s running mate last year, said on Tuesday. “This is moving into perjury, false statements and even potentially treason.”

Republicans in Congress made little effort to defend the White House, and some expressed concern. “I voted for @POTUS last Nov. & want him & USA to succeed, but that meeting, given that email chain just released, is a big no-no,” Representative Lee Zeldin, Republican of New York, wrote on Twitter, using the acronym for president of the United States.

The White House rejected the criticism. Asked about the use of the term “treason,” Sarah Huckabee Sanders, the principal deputy White House press secretary, said, “I think those new words are ridiculous.”

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Mr. Trump’s son and the president’s defenders said that once again, the fuss was unwarranted. “Media & Dems are extremely invested in the Russia story,” Donald Trump Jr. wrote on Twitter. “If this nonsense meeting is all they have after a yr, I understand the desperation!”

Other defenders of Mr. Trump have sought to minimize his campaign’s dealings with Ms. Veselnitskaya by arguing that they are similar to the interactions between Ukrainian government officials and a Democratic Party consultant who sought damaging information about Mr. Trump and his associates, including Mr. Manafort, during the campaign.

On a recent program, the Fox News host Sean Hannity, who is close to Mr. Trump, read from a January article in Politico revealing the Democratic operative’s meetings with Ukrainian officials in the country’s Washington embassy. “Where is the outrage over this?” Mr. Hannity demanded.

Inside Mr. Trump’s team, the response to the developments has become a subject of intense dispute. Three people familiar with his son’s account said he pushed to offer a full explanation on Saturday when first contacted by The Times about the meeting, and said he agitated to be allowed to defend himself publicly. But three other people involved in the discussions offered a completely contrary version of events, insisting that the younger Mr. Trump adamantly resisted an expansive disclosure.

Elsewhere, Mr. Kasowitz was working separately to inform an article being prepared by Circa, a news outlet the White House considers favorable, instead of The Times. According to the person close to the legal team, Mr. Kasowitz was kept out of the discussion about Donald Trump Jr.’s initial statement and saw it only after it had been published online in the first Times article.

The original statement, drafted aboard Air Force One by advisers and then approved by Mr. Trump, said only that the Russian lawyer had discussed adoption policy during the meeting, without mentioning that the meeting had been offered as a chance to provide information about Mrs. Clinton’s dealings with Russia. Only after The Times followed up in preparation for another article did the younger Mr. Trump issue a second statement acknowledging that.

On Tuesday, the Trumps projected a united front.

“This is the EXACT reason they viciously attack our family!” Eric Trump, the president’s second-oldest son, posted on Twitter. “They can’t stand that we are extremely close and will ALWAYS support each other.”