WASHINGTON — Since Donald Trump called for temporarily banning Muslims from entering the U.S., he has tried to expand, narrow or otherwise redefine the polarizing proposal that helped him win the Republican primary but has posed a greater challenge in the general election campaign.

On Monday, he added a phrase to his policy lexicon: “extreme vetting.”

To Trump, that means ensuring anyone entering the country shares American values.

The newest addition to Trump's immigration policy came during a major Youngstown, Ohio, speech on national security that featured an unusually subdued Trump reading uneasily at times from a teleprompter and repeating several false claims, including his assertion that he was an early opponent of the Iraq invasion.

The speech followed days of criticism over Trump's insistence that President Barack Obama and Democratic nominee Hillary Clinton founded the Islamic State. Those comments, and other unscripted and unforced controversies, have helped distract from Trump's core economic and anti-terrorism messages, push down his standing in polls and lead some fellow Republicans to again urge him to curtail his style of campaigning.

Trump did not explicitly back down from his December proposal for “a total and complete shutdown of Muslims entering the United States until our country's representatives can figure out what is going on.”

He did not mention it, instead calling on the departments of State and Homeland Security “to identify a list of regions where adequate screening cannot take place,” leading to a temporary visa halt.

Trump spent more of his speech defining what he said was a new ideological test for those entering the U.S., comparing his plan to Cold War-era screening.

“We should only admit into this country those who share our values and respect our people,” he said. “In addition to screening out all members or sympathizers of terrorist groups, we must also screen out any who have hostile attitudes towards our country or its principles — or who believe that Sharia law should supplant American law. Those who do not believe in our Constitution, or who support bigotry and hatred, will not be admitted.”

The newest iteration of Trump's policy, while not specifically demanding a religious test for entering the country, still allows for capricious enforcement, said Steve Yale-Loehr, a Cornell Law School professor who specializes in immigration.

“What one president thinks is important for American values, another president may deem not important,” he said. “We don't want an immigration policy subject to the vagaries of political opinion.”

State Department spokeswoman Elizabeth Trudeau declined to comment on Trump's immigration proposals but said: “We stand by the integrity of our visa process.”

Trump on Monday mostly delivered broad outlines for his ideas on fighting terrorism, rather than specific policy proposals. Some ideas, like relying on more human intelligence to target terrorists in addition to drone strikes, echo Obama administration policy. The message from Trump, however, was that Obama and Clinton have tiptoed around the threat because they are unwilling to use the phrase “radical Islamic terrorism” and are too afraid of offending those who would do harm to effectively target them.

Though his call to ban Muslims has drawn accusations that he is fomenting bigotry, Trump said his policies were instead geared toward national unity and fighting an ideology that promotes oppression of women and gays. He called for sending home those who preach hate.

At the same time, he cast suspicion also on second-generation immigrants, saying their status, along with those born in other countries, was a common thread in several terrorist attacks.

Trump, who has vacillated in recent days on his charge that Obama and Clinton were the founders of the Islamic State, also known as ISIS, attempted to modify that assertion Monday. Instead of again calling them the literal founders, he said “the rise of ISIS is the result of policy decisions made by President Obama and Secretary of State Clinton.” He singled out the withdrawal from Iraq.

Trump's speech came amid increasing levels of controversy surrounding his campaign. His campaign chairman, Paul Manafort, denied a New York Times story that told of handwritten ledgers indicating he received $12.7 million in undisclosed cash payments from a pro-Russian political party in Ukraine.

Washington Bureau's Michael A. Memoli in Scranton, Pa., and Joseph Tanfani and Tracy Wilkinson contributed.

noah.bierman@latimes.com