An Israeli military spokesman, Lt. Col. Nadav Shoshani, said on social media the rockets had hit a civilian neighborhood that was not near a military target. “Attacks against our civilians will not go unanswered,” he said

Magen David Adom, the Israeli emergency medical service, said a 30-year-man in Katzrin had been treated for shrapnel wounds and was taken to a hospital.

The strikes by both sides, and the Israeli official’s threat of further retaliation, highlighted how months of diplomatic efforts have failed to ease hostilities along the Israel-Lebanon border. And they came as the Biden administration has intensified its push for a cease-fire between Israel and Hamas in Gaza in hopes of averting a broader Middle East war, although Israeli and Hamas officials have been cool to the latest U.S. proposal.

The most recent Israeli strikes landed about 40 to 60 miles north of the Israel-Lebanon border in the Bekaa Valley, deeper inside Lebanon than many of the near-daily attacks the two countries have exchanged since the war in Gaza began. Hezbollah, like other groups in the region backed by Iran, has been attacking Israel in solidarity with the Palestinians in Gaza. Hamas is also backed by Iran.

The military said it had detected secondary explosions after its strikes on Wednesday, which it said indicated that there were large amounts of weapons at the sites. At least three areas were targeted, including the town of Nabi Chit, according to Lebanon’s state-run news agency.

On Monday, the Israeli military also said it had targeted a number of Hezbollah’s weapons storage facilities in the Bekaa Valley. Lebanon’s Health Ministry said those strikes injured nearly a dozen people, including two children.

Bachir Khodr, the governor of the region where the strikes hit, said he could not comment on Israel’s targets but that residential homes had been damaged.

Mr. Khodr said that local officials had met after the strikes and were preparing for the possibility of a wider war, though people in the area were not in “panic mode.”

“We hope it doesn’t happen, but in case it does, we are ready,” said Mr. Khodr, who added that the authorities had made plans to relocate civilians, distribute food and water and maintain health services.

In a separate strike on Wednesday, the Israeli military said it had killed a commander in the Aqsa Martyrs Brigades, a militant group associated with the Palestinian Fatah faction that has fought alongside Hezbollah. The commander, Khalil al-Miqdah, who was killed in the strike in the southern Lebanese port city of Sidon, worked closely with Iran’s powerful Revolutionary Guards, the Israeli military said in a statement. That claim could not be independently verified, though the Aqsa Martyrs Brigades confirmed Mr. al-Miqdah’s death in a statement.

The attack on Katzrin involved numerous rockets, most of which did not get through the air defenses, the Israeli military said. A few did, shattering the morning. Brig. Gen. Alon Friedman, a spokesman, said the rockets had hit two houses as their occupants were sitting down to breakfast.

Inbar Kedem, who comes from Kibbutz Be’eri in southern Israel, was on a camping trip with friends in the Golan Heights near Katzrin when he saw some of the rockets being intercepted by air defenses. He said such events were routine in Israel now.

“This is a classic day in the war,” said Mr. Kedem, who was just discharged from the Israeli Air Force. “Just a little more intense.”

On Wednesday, Ms. Woodruff released a statement calling her reference to the reports, which she said were from both Axios and Reuters, “a mistake.”

“In the live TV moment, I repeated the story because I hadn’t seen later reporting that both sides denied it,” her statement said. “This was a mistake, and I apologize for it.”

Mr. Trump and Mr. Netanyahu last met on July 26 at Mr. Trump’s club, Mar-a-Lago, in Palm Beach, Fla., in an effort to smooth out their relationship, which was strained after Mr. Netanyahu congratulated President Biden on winning the 2020 election. (Mr. Trump has never conceded he lost.)

Mr. Trump’s campaign spokesman said the two men had not discussed slowing down a possible cease-fire agreement.

“The only thing President Trump has told the prime minister is ‘to get the war over with,’” said Steven Cheung, the Trump campaign communications director. “Any assertion otherwise is fake news.”

A spokesman for Mr. Netanyahu, Omer Dostri, said that reports that Mr. Trump had asked the prime minister to wait on a deal were a “total lie” and it never happened.

Axios reported last week that the two had discussed the cease-fire in a phone call on Aug. 24, citing two unnamed “U.S. sources.” One of the sources said that Mr. Trump’s call was intended to encourage Mr. Netanyahu to take the deal. Reuters published a report about the Axios account.

Mr. Netanyahu’s office quickly denied the call had taken place and Axios updated its story. A person close to Mr. Trump also denied that the two men had spoken since the July 26 meeting.

The Israeli police said that they had sent reinforcements to try to maintain order, and Israeli news media reported that officers had sealed off several streets, used water cannons to disperse crowds and beaten some protesters with batons. When asked about the response, the police said in a statement that officers had been “forced to act using various means” as protests continued and demonstrators broke through a blockade, with some protesters throwing water bottles. Five people were arrested, the police statement added.

The protest highlights the increased friction between Israel’s mainstream secular society and the ultra-Orthodox, the fastest-growing part of the population.

Some ultra-Orthodox Israelis do not fully recognize the state of Israel, rejecting secular Jewish sovereignty and military service. Many ultra-Orthodox see full-time Torah study as crucial, arguing that this scholarship is what has ensured the survival of Jews for centuries.

A debate long seen by the rest of Israeli society as one over equality has increasingly become one about security, too. Israel has been in a 10-month war with Hamas in Gaza, and skirmishes with the Lebanese militant group Hezbollah have intensified. Fear of a regional war still looms amid concerns that both Hezbollah and its patron, Iran, could launch retaliatory attacks for recent assassinations attributed to Israel.

A video from the Israeli broadcaster Channel 7 showed of one of the altercations at the protests on Wednesday, with one ultra-Orthodox demonstrator asking a counterprotester, “You want me to work for you?”

“Your protection is worthless,” the counterprotester says, in an apparent reference to the Torah. He soon punches the ultra-Orthodox man.

Israel’s military hoped to defuse tensions over the conscriptions of the ultra-Orthodox — some 4,800 are to be drafted this year — by focusing on unmarried male members of the community who are in the work force and not on religious students.

But last week, hundreds of ultra-Orthodox protested outside the Alon military base. Out of 90 ultra-Orthodox candidates for service who were summoned to the base that day, only 12 showed up and completed the processes required for conscription.

Tensions over the draft pose yet another test for the government of Prime Minister Benjamin Netanyahu, who has struggled to balance the demands of the ultra-Orthodox parties that form a critical part of his coalition with his own nationalist base, some of whom no longer believe the exemptions are viable.

Negotiations mediated by the United States, Egypt and Qatar have inched along since early December, when hostilities resumed after a one-week cease-fire during which Hamas released more than 100 people from captivity in Gaza and Israel freed 240 Palestinian prisoners. In late May, President Biden endorsed a new three-phase plan and the U.N. Security Council followed with a resolution supporting it.

The first phase would see a six-week cease-fire and the release of hostages in exchange for Palestinians being held in Israeli prisons. People displaced from northern Gaza would be able to return to their homes, many of which lie in ruins. During that time, Israeli forces would withdraw from populated areas of Gaza.

The second phase envisions a permanent cease-fire, while the third consists of a multiyear reconstruction plan for Gaza and the return of the remains of deceased hostages.

But for months, Israel and Hamas, whose negotiators do not speak directly to each other, have remained far apart on key issues.

What is the new U.S. proposal?

On Aug. 8, with the war in its 11th month, President Biden and the leaders of Egypt and Qatar said they were willing to present a “final” cease-fire proposal. Last week, at talks in Qatar, the United States presented what it called a “bridging proposal” to try to close some of the gaps between Israel and Hamas.

The details of that proposal have not been disclosed publicly, but the Biden administration has tried to put diplomatic heft behind it. Visiting Israel this week, Secretary of State Antony J. Blinken warned that this was “maybe the last opportunity” to secure a cease-fire, and later said that Prime Minister Benjamin Netanyahu had told him during a meeting that he was committed to the U.S. proposal.

But both Israeli and Hamas officials familiar with the talks said the U.S. proposal leaves major disagreements mostly unresolved.

What is the main sticking point?

In broad terms, the U.S. proposal appears to conform to new demands added by Mr. Netanyahu in July that some Israeli troops continue to patrol part of an area of Gaza along the border with Egypt, according to Hamas and Israeli officials.

This has emerged as a crucial issue. Mr. Netanyahu considers an Israeli military presence in the area, which Israel calls the Philadelphi Corridor and Egypt calls Salah Al Din, vital to preventing Hamas from rearming after the war or rebuilding tunnels to Egypt.

Mr. Netanyahu told a group, including families of hostages, this week that Israel would not withdraw from the border strip “under any circumstances,” the families said in a statement. Mr. Netanyahu’s office confirmed he had said that Israel would not withdraw from Philadelphi.

Hamas rejects a continued Israeli presence in the area and is demanding a complete Israeli withdrawal from Gaza. Egypt says that keeping Israeli troops in the Philadelphi Corridor would raise national security concerns and would be unacceptable to the Egyptian public.

Mr. Blinken told reporters on Tuesday that Israel has already agreed to terms of withdrawal and reaffirmed that the United States would “not accept any long-term occupation of Gaza by Israel.”

What are the other disputes?

Other areas of dispute have emerged publicly. Since the start of the war, Israeli forces have established what it calls a security buffer inside Gaza along its eastern border with Israel, demolishing Palestinian homes in the process, and it intends to keep a presence there after the war. Israel also wants to retain the option to return to fighting after the first phase of a cease-fire.

Israeli forces have built a security road, which they call the Netzarim corridor, that cuts across Gaza from east to west. Israeli officials have said they want troops to keep patrolling that road, through which Palestinians must travel between the north and south of the enclave. That would violate Hamas’s insistence on a complete Israeli withdrawal.

The lineup of speakers that convention organizers released on Wednesday evening included Mr. Polin’s parents, Rachel Goldberg and Jon Polin.

Uncommitted’s campaign to make the war in Gaza a focus of the convention comes as the death toll there has surpassed 40,000, according to Gazan health authorities. Hopes for the survival or recovery of the more than 100 hostages, some believed to be dead, who are still being held in the enclave have grown dimmer. The Israeli military announced Tuesday that its forces had recovered the bodies of six hostages from a tunnel in Gaza, and the latest effort by the Biden administration to negotiate a cease-fire between Israel and Hamas appeared to have reached a stumbling block, with many issues as yet unresolved.

Margaret DeReus, who heads the Middle East Institute for Understanding, a nonprofit advocacy group working with Uncommitted, said in a phone interview from Chicago that as she and her colleagues have walked the convention floor — wearing stoles printed to resemble kaffiyehs, the black-and-white scarves that have become a symbol of Palestinians, and bearing the words “Democrats for Palestinians” — many attendees have expressed strong support for a permanent cease-fire. But the official convention program has left her feeling “completely alienated,” she said, giving her “a feeling that the party wants to brush the issue of Gaza under the rug.”

Democratic leadership is out of touch with its base when it comes to the war in Gaza, Ms. DeReus argued. But she said she is hopeful that Vice President Kamala Harris, Democrats’ presidential hope, will “say something that indicates a serious, tangible shift” from the Biden administration’s current policies when she accepts the party’s nomination on Thursday.

Ms. Harris has expressed “more empathy for Palestinian suffering” than the president, she said, but “what we need to hear is that her administration will be different.”