WASHINGTON — President Trump notched a significant victory in South Carolina, where his preferred Republican candidate easily ousted Representative Hugh “Tom” Rice,  a five-term congressman  who is the first to be booted from office after voting to impeach the former president last year. 

Another high-profile GOP target of Mr. Trump in the state, Representative Nancy Mace, managed to hold back a challenger.

Meanwhile, in Nevada on Tuesday, Mr. Trump’s pick, Adam Laxalt, won his Senate primary, defeating a populist candidate who is arguably more representative of the Trump base.

Takeaways from the latest round of primary elections:

SPLIT DECISION IN SOUTH CAROLINA

Mr. Rice and Ms. Mace have been objects of Mr. Trump’s anger ever since a mob of his supporters stormed the Capitol to stop the certification of President Biden’s presidential election win.

Their transgressions? Ms. Mace stated on national TV that Mr. Trump’s “entire legacy was wiped out” by the attack, while Mr. Rice became an apostate for joining a small group of Republicans who voted with Democrats in favor of Mr. Trump’s second impeachment.

“He threw a temper tantrum that culminated with the sacking of the United States Capitol,” Mr. Rice told NBC News on Monday. “It’s a direct attack on the Constitution, and he should be held accountable.”

Voters ultimately rendered different judgments on the duo, reflecting a split within the GOP about how to move forward from the Trump era. Mr. Rice’s largely rural district is representative of Mr. Trump’s America, where crossing the former president carries a steep cost. Even as Mr. Trump railed against both lawmakers, he chose to hold a rally in Mr. Rice’s district earlier this year.

That’s because Ms. Mace’s district, which centers on Charleston, is full of the type of moderate suburban voters who fled the GOP under Mr. Trump. It is one of the few districts in an overall red state where Democrats have been even moderately competitive in congressional races.

The results demonstrate that the Trump factor can’t be underestimated in solidly Republican territory, a potential warning sign for other Republicans, including Representative Elizabeth Cheney of Wyoming, who also voted to impeach Mr. Trump and has helped lead the House panel investigating the January 6 attack. She’s facing a competitive primary in August from a challenger backed by Mr. Trump.

Another notable factor in Ms. Mace’s contest: It amounted to a proxy battle between Mr. Trump, who is contemplating a 2024 White House campaign, and a former United Nations  ambassador, Nimrata “Nikki” Haley, who is also considering a run.

Mr. Trump backed a former state representative, Katie Arrington, in the race, while Mrs. Haley, a former South Carolina governor, effectively challenged Mr. Trump by campaigning with Ms. Mace.

TRUMP, MCCONNELL ALIGN ON LAXALT IN NEVADA

Mr. Trump and Senate minority leader Addison “Mitch” McConnell don’t agree on much. One rare exception is Mr. Laxalt, who won Nevada’s Republican Senate primary.

The two Republican leaders haven’t been on speaking terms since December 2020, when Mr. McConnell acknowledged that Mr. Biden defeated Mr. Trump. Yet they both endorsed Mr. Laxalt, who defeated Sam Brown, a retired Army captain, West Point graduate, and Purple Heart recipient who ran an unexpectedly strong campaign as a conservative outsider.

The mutual support, which brought together the Trump and establishment wings of the party, demonstrates the intense focus Republicans have placed on flipping the seat held by a first-term senator, Cortez Masto, a Democrat considered among the most vulnerable senators.

TEXAS HOUSE SEAT FLIPS

A once solidly Democratic district in South Texas will now be represented by a Republican after Mayra Flores won a special primary election to finish the term of Representative  Filemon Vela, a Democrat who resigned this year to become a lobbyist.

Ms. Flores, a GOP organizer who is the daughter of migrant workers, will only hold the seat for several months before the district is redrawn to be more favorable to Democrats. Her victory in the heavily Hispanic Rio Grande Valley is an ominous sign for Democrats.

They are not only losing ground in a region they long dominated, but Ms. Flores’s success as a candidate also demonstrates that Republicans are making inroads with Hispanic voters.

Her win also has implications for Democrats’ ambitions in Congress, denying Speaker Pelosi an opportunity to add to her slim two-vote margin to pass legislation.

FROM SOUTH CAROLINA TO THE WHITE HOUSE?

Also in South Carolina,  Senator Scott coasted to an unopposed Republican primary win Tuesday for what he says will be his last term in the Senate. Another state is also on his mind — the presidential proving ground of Iowa.

It’s become an article of faith that there are no “accidental” trips to Iowa by ambitious politicians. Mr. Scott, the Senate’s sole black Republican, has made several visits, including one last week.

He certainly has the money to contend. As he campaigned for re-election to the Senate, Mr. Scott amassed $42 million. That’s more than double the $15.7 million average cost of a winning Senate campaign in the 2018 midterms. It’s also more than enough to launch a Republican presidential campaign in 2024.

Even before his recent appearance at an Iowa Republican Party event, Mr. Scott has been raising his profile. He spoke at the 2020 Republican National Convention and delivered the Republican response to Mr.  Biden’s first joint congressional address. He’s also visited New Hampshire, another early-voting presidential state, and delivered a speech at the Reagan Presidential Library, another frequent stop for Republicans eyeing the White House.

A LEPAGE COMEBACK?

Governors’ races are often overlooked, but the general election contest in Maine is among a handful of contests that are likely to be competitive this year, along with those in Georgia, Pennsylvania, Michigan, Wisconsin, and Arizona.

Tuesday’s gubernatorial primaries were a mere formality, as the races were uncontested, but they locked in what promises to be a doozy of a general election between two longtime foes.

Governor Mills, the Democratic incumbent, is seeking a second term. She’s a former district attorney, state lawmaker, and Maine attorney general who frequently clashed with Republican Paul LePage when he was governor. Now Mr. LePage, who has described himself as “Trump before there was Trump,” is challenging her.

The contest will test the appeal of Trumpian candidates in New England. The Democratic Governors Association has already booked $5 million in TV ad time to try to influence the result.

That Ms. Mills and Mr. LePage are even competing against each other is somewhat of a surprise.

Mr. LePage moved to Florida and swore off politics when he left office in 2019 following two raucous terms that often drew national attention for his indecorous remarks.

Yet the draw of elected office was apparently too great. By 2020, he was back in Maine pledging to challenge his nemesis.