The deep distrust that has marked American-Israeli relations during most of the Carter administration lessened perceptibly yesterday as Israeli Prime Minister Menachem Begin repeatedly assured Secretary of State Cyrus Vance that Israel would work with him to get peace talks with Egypt resumed.

Israeli leaders appear to remain wary of the Carter administration's intentions in the Middle East, but they are saying in a variety of ways during Vance's visit that they do not see pressure from the White House as an immediate threat to them.

This new confidence grows laregly out of a perception here that Egyptian President Anwar Sadat has sacrificed his image as a peacemaker in world opinion and particularly in the United States by ordering all direct talks and links with Israel cut off on July 30. Concern that the Sadat initiative is dying is coupled here with relief that Israel can avoid being blamed for its demise.

In their public remarks since Vance arrived Saturday night, Begin and Foreign Minister Moshe Dayan have been careful to nurture the view that Israel is cooperating completely with Vance's effort to revive the Sadat peace initiative and that the obstacle to peace now is Egypt.

"If there will be an atmosphere in Alexandria as there was in Jerusalem, then he [Vance] will succeed," Begin said after ending four and a half hours of talks with the secretary yesterday Vance travels to the Egyptian port city today to see Sadat.

Israeli leaders "are not crowing about it, but they do think the pressure on them is off as a result of Sadat breaking off talks," an American official said. "They see a change in American public opinion, and feel more comfortable because of it."

The cordial Israeli reactions to the talks indicated that Vance had not attempted to press Israel into any new concessions designed to get Sadat back to the bargaining table at this point, and that he did not bring an "American peace plan" that Sadat has called for to break the deadlock.

Begin sounded euphoric last night when he said his meeting with Vance was "the best we have ever had since the secretary's five trips to the Middle East. "There was no American request to Israel to change its position," Begin added.

Begin seemed to be trying hard to avoid appearing to be negative about anything he discussed with Vance. He commented about a handwritten note that Vance brought to him from President Carter in friendly and positive tones. Both sides declined to discuss details of the letter or of yesterday's talks.

Asked by reporters about unconfirmed news reports to an American proposal for a summit meeting in Washington that would bring Begin, Carter and Sadat together to break the deadlock, Begin said the idea did not come up during his talks with Vance. If it were proposed later, "I would consider it with great seriousness," he added. Israel has traditionally favored the kind of direct two Party talks initiated by Sadat's visit to Jerusalem last November.

Begin, who appeared relaxed and joked with reporters, turned the other cheek to what he called "the totally negative Egyptian statements" on new talks and personal attacks against him by the Egyptian media. "I do not react because I do not want to exacerbate the situation," Begin said calmly.

Nearly half of the 2 1/2-hour morning session centered on southern Christian militiamen have refused to let a Lebanese army battalion move into the militia-controlled buffer zone on the Israeli-Lebanese border.

Vance, after meeting with Begin privately for 20 minutes, heard Israeli Cabinet members and military commanders defend their actions in southern Lebanon. The secretary was believed to have emphasized U.S. hopes that Israel would cooperate in getting Lebanese army units formally reestablished in the area.

One sign of a lessening in tension between Washington and Jerusalem, however, temporary, came in an analytical article published in yesterday's Jerusalem Post. The article was written by Salman Shoval, a member of Parliament who is also head of the Foreign Ministry's advisory committee on information activities abroad and who is close to Dayan. Shoval is known to have written the article after its principal themes were discussed at a senior staff meeting Friday and approved by Dayan.

"President Sadat has become a victim of his own public relations ploy." Shoval wrote. After convincing the world he wanted peace. "Sadat has now rejected the Israeli proposals lock, stock and barrel."

"Israel is doing its best to keep the peace process moving," he said.

Sadat believes "that the U.S. can and will put pressure on Israel" and has created an atmosphere of crisis "to force America's hand and make her propose a plan of her own" Shoval asserted. "For once, Egypt's master of public relations seems to have miscalculated" and triggered negative responses not only from the U.S. and Western European press, but also from the Carter Administration itself," he said.

Other Israeli newspapers reported yesterday that Israel would put forward no new proposals during the talks with Vance and that the Americans would not present their own peace plan.

Dayan also alluded to the Carter administration's approach to the Middle East without bitterness in a television interview Friday night. "One thing I really didn't like was placing Israel in the accused's dock in a persistent manner of a numer of months and portraying Israel as hard-lined," the foreign minister said.

"But the Americans have never actively taken one side or the other" he continued, and have not "forced us to accept their plan, which they have and publicized . . . So I have no argument with the way the Americans are carrying out their mediation efforts."

Dayan specified that President Carter in his first months in office had publicly described a settlement that envisioned Israeli withdrawal to the borders of 1967 with minor adjustments and Palestinian homeland.

Sadat's refusal to go forward with new direct talks followed a mid-July meeting of Vance, Dayan and Egyptian Foreign Minister Ibrahim Mohammed Kamel's at Leeds Castle in England, where Dayan said Israel might be willing to discuss returning part of the West Bank and Gaza Strip territories to Arab sovereignty after an interim 5-year period of self-rule adminstered under continued Israeli occupation.

Dayan apparently specified that the only possible outcomes of such future negotiations would be Jordanian, Isracli or joint Jordanian-Israeli sovereignty, or continued occupation. This would rule out an independent Palestinian state.

The carter administration views the Israeli proposals as primarily tactical concessions at the moment, because Begin is believed by the White House to be still committed to his Biblical vision of the West Bank as part of sovereign Israeli territory. But Sadat's cancellation of talks, described by the State Department as "deeply disappointing," has eclipsed discussion of them.