Tuesday was a resounding rejection of former President Donald J. Trump’s obsessive focus on the 2020 election. In Georgia, his choices for governor and attorney general were soundly defeated, while his choice for secretary of state was defeated by a man who stood up to false claims two years ago.

However, it would be a mistake to interpret these results as a blanket rejection of Mr. Trump. His gravitational pull on Republican voters skewed every primary on Tuesday, shaping candidates’ positions and priorities as they raced to Mar-a-Lago.

It was a bittersweet evening for progressives in Texas, who are still waiting to hear the fate of their challenger to a conservative Democratic incumbent. In another House race, the party’s left flank defeated one of the “unbreakable nine” Democrats who opposed President Biden’s social spending plans in the Atlanta suburbs.

Here are a few key takeaways from this week’s primaries, which are among the most important of the 2022 midterm election cycle:

In the final days of the race, David Perdue, a wealthy former senator recruited by Mr. Trump to challenge Georgia Gov. Brian Kemp, told reporters, “I guaran-damn-tee you we’re not down 30 points.”

Mr. Perdue was right. He was defeated by about 50 percentage points.

Mr. Kemp easily defeated Mr. Perdue’s lackluster bid, rallying local support and fellow Republican governors to his cause. By the final weeks of the campaign, Mr. Perdue had reduced his reliance on television advertising, which is usually a sign of a doomed candidacy.

Even though Mr. Trump had transferred more than $2.5 million from his political operation to Mr. Perdue, it wasn’t enough. Mr. Perdue’s own supporters were openly critical of his sluggish campaigning, as well as his inability to move past false claims about the 2020 election.

Republican governors rushed to characterize Mr. Kemp’s landslide victory as a rejection of Mr. Trump. Minutes after Mr. Perdue conceded, Chris Christie, the former governor of New Jersey and a Trump supporter, praised Georgia voters for refusing to be “willing participants in the DJT Vendetta Tour.”

Mr. Trump’s participation can also tilt an entire primary contest to the right, as it did in Alabama and Georgia. Mr. Kemp now faces a rematch in the general election against Stacey Abrams, an experienced and well-funded Democrat whom he defeated in 2018 by fewer than 55,000 votes.

Mr. Trump’s record in actual contested primaries has been more mixed than his overall win-loss record suggests.

His preferred Senate candidates were successful in Georgia, North Carolina, and Ohio, but struggled in Alabama and Pennsylvania.

In governor’s races, he backed Sarah Huckabee Sanders, his first White House press secretary, who easily won in Arkansas, where she is political royalty. Mr. Trump was critical of Alabama Gov. Kay Ivey on occasion, but she avoided a runoff in her primary.

But he also ran unsuccessfully against Republican incumbents in Georgia and Idaho, and his choice for governor of Nebraska, Charles Herbster, was defeated by Jim Pillen, the local establishment’s favorite, by nearly four percentage points this month.

Candidates who used Mr. Trump’s story of a rigged election as the centerpiece of their campaigns fared poorly. Those who only partially embraced it, on the other hand, fared well.

Brad Raffensperger won the Republican primary for Georgia Secretary of State by an outright victory over Representative Jody Hice, whose unwavering support for Mr. Trump’s conspiracy theories about the 2020 election was insufficient to force a runoff.

Chris Carr, the Republican nominee for attorney general, brushed aside a feeble challenge from John Gordon, a lawyer who represented Mr. Trump’s bogus election-fraud claims in court. Mr. Raffensperger may have received assistance from Democrats, thousands of whom reportedly voted Republican.

“Not bucking under pressure is what the people want,” Mr. Raffensperger said at his election watch party on Tuesday night.

However, few Republican candidates who have openly denounced Mr. Trump’s lies about 2020 have fared well elsewhere.

In Ohio, the one Senate candidate who did so, Matt Dolan, finished in third place. In Pennsylvania, the Republican nominee for governor, Doug Mastriano, was deeply involved in Mr. Trump’s plot to overturn the state’s 2020 results, while the two leading Senate candidates, Dr. Mehmet Oz and David McCormick, have equivocated about whether Mr. Biden was fairly elected.