JERUSALEM — After months of equivocation, Israel has formally recognized last year’s offensive against Lebanese Hezbollah guerrillas as a war and given it a name.

But while the chosen title — “Second Lebanon War” — may offer some comfort to Israelis who were bereaved by the fighting and want to know their losses were for a cause, it’s unlikely to help the embattled Olmert government.

Designating the conflict as a war could fuel public ire at Israel’s perceived hesitation to authorize a full-bore military assault after Hezbollah abducted two soldiers in a deadly July 12 border raid.

Instead, Prime Minister Ehud Olmert and Defense Minister Amir Peretz at first relied on air force and artillery shelling, which exacted a steep price on Lebanon’s civilian infrastructure and drew international censure while leaving Hezbollah to launch thousands of short-range rockets into northern Israel. By the time troops and tanks were finally unleashed on Hezbollah’s frontier strongholds, days before a U.N.-brokered ceasefire on Aug. 12, the damage to Israel’s prestige was done.

Olmert and Peretz, sensing domestic support shifting toward the right-wing opposition, have insisted that the war improved national security by restoring Israel’s military deterrence and boosting a foreign peacekeeper force and regular Lebanese troops in Hezbollah’s former heartland.

“When they try to surface now they are disarmed and arrested by the international force and the Lebanese force,” Olmert told reporters last month, referring to Hezbollah fighters. “I am not certain that they have any appetite to fight with Israel again.”

But with ample reports that Hezbollah is rearming and that Israel’s tactical setbacks emboldened its arch-enemies Iran and Syria, Olmert has had a hard time convincing his countrymen.

A recent poll found that if elections were held now, Olmert would take only 3 percent of votes. His top rival, the hawkish former Prime Minister Benjamin Netanyahu, appears to be the frontrunner.

Olmert’s fate may hinge on a committee of inquiry set up to investigate the war. The so-called Winograd Committee is expected to issue interim results next month, including conclusions about Olmert’s and Peretz’s responsibility for the campaign.

Under a High Court of Justice order, the Winograd Committee on Thursday published some of the testimony it took from high-level Israeli officials. One segment included an implicit critique of Olmert and Peretz by Vice Premier Shimon Peres, Israel’s elder statesman.

“Had it been up to me, I wouldn’t have gotten into this war, ” Peres said, adding that the military had not been properly prepared. Peres further rapped Olmert for declaring, early on in the campaign, that he would win the return of the two captive soldiers. This, Peres said, put Israel “at the mercy of the enemy. “

For now, naming the Lebanon war answers the demands of relatives of the 117 Israeli soldiers and 41 civilians killed by Hezbollah gunmen and rocket crews.

The temporary epitaphs on the graves of the dead troops read: “Fell in combat in southern Lebanon “. That stoked concern that the government might try to play down its role in the conflict’s failings by designating it as less than a “war. “

During the fighting, Israel made do with military codes for the campaign — “Operation Just Reward ” and “Operation Change of Direction. ” But Cabinet Minister Yaacov Edri, who heads the government’s Protocol and Ceremonies Committee, announced the new name Wednesday. It is expected to be ratified next week.

“Second Lebanon War ” refers implicitly to Israel’s 1982 invasion of Lebanon in an offensive against the Palestine Liberation Organization. Israel did not officially call that campaign a war, but referred to it as “Operation Peace for Galilee. “

Edri denied the choice of name posed problems for posterity.

“It’s a declarative matter, without any formal or legal ramifications, ” he told Israel’s Channel Ten television. “It’s also a catchy name and there’s no dispute about it. “

Peace for Galilee led to the expulsion of Yasser Arafat and the rest of the PLO’s leadership from its Lebanon base. Just a decade later, Arafat would become Israel’s partner in the ill-starred Oslo peace accords.

Meanwhile, Israeli forces remained in southern Lebanon after the 1982 conflict, creating a security zone which, in turn, gave rise to Hezbollah.