Miami lawyer Richard Stone, a former Democratic senator from Florida, will resurface soon as public spokesman for the Reagan administration's policy in the Caribbean and Central America.

Under the rubric of the administration's new "public diplomacy" offensive, Stone will defend continued U.S. backing for the government of El Salvador and the Caribbean Basin Initiative, a congressional casualty last year. His designation as point man for this policy strengthens the administration's stance of bipartisanship but reportedly has upset Assistant Secretary of State Thomas O. Enders, frequently the spokesman on Central American affairs.

The administration is going all-out on the new initiative, seeking to develop what one official calls "a more effective explanation of foreign policy objectives."

The public diplomacy effort will place heavy reliance on a speakers bureau of high-ranking administration officials who will give the Reagan pitch on Soviet policies, arms control, the Middle East and Central America. Last week, national security adviser William P. Clark encouraged members of the National Security Council staff to use off-duty hours to explain administration policies to their church, service and fraternal groups.

The effort is being spurred by tardy White House recognition that Reagan policies, particularly on arms control, are on the defensive throughout the world in the face of an effective campaign from the new Soviet leadership.

"We are losing the public relations battle," warned Charles Z. Wick, director of the U.S. Information Agency, in a recent interview. Wick, a close friend of the president, has been urging a stepped-up public relations effort for more than a year.

Wick said the $65 million proposed in the Reagan budget for "democracy initiative" projects will not be used for the public diplomacy campaign, which is funded from existing agency budgets. The $65 million is earmarked for long-range efforts to promote understanding of U.S. views through scholarships, exchange programs and support of democratic political groups and labor unions.

Word is out in the White House that Rich Williamson, presidential assistant for intergovernmental relations, will soon be on his way to Vienna to serve in a new ambassador-level administrative post to United Nations international organizations. Williamson has the backing of chief of staff James A. Baker III and deputy chief of staff Michael K. Deaver for the job.

Baker and former presidential political adviser Lyn Nofziger attempted to patch up their differences recently with a luncheon at a downtown restaurant where both agreed that the president's reelection campaign, when and if it comes, should be managed by an outside committee. Nofziger picked up the tab.

It was their first get-together since Baker was quoted in a Dallas newspaper a month ago expressing satisfaction about his successful attempts in December to nip in the bud a meeting of former Reaganites who wanted to plan the 1984 campaign. "We cut him off at the knees," Baker was quoted as saying of Nofziger at the time.

John Herrington's return to the White House personnel office where he served as troubleshooter more than a year ago prompted one administration official to recall one of Reagan's oft-told stories about the tenor who sings an aria, receives a prolonged ovation and repeats it as an encore.

This happens several times until the tenor pleads weariness and says he must stop singing encores or his voice will give out.

"You'll do it until you get it right," cries a voice from the crowd. "And so," adds the administration official, "will John."

Next entry on the Reagan book list is "Gambling With History: Reagan in the White House" by Laurence I. Barrett, a White House correspondent for Time magazine. It is scheduled for publication by Doubleday June 24. Barrett predicts that Reagan, if his health holds up, will seek a second term.

It used to be said that there are no atheists in foxholes, and they apparently also are in short supply during recessions. Beginning his speech to a convention of religious broadcasters last week, our optimistic president had these words of cheer: "In a time when recession has gripped our land, your industry, religious broadcasting, has enjoyed phenomenal growth." Reaganisms of the Week: The president, to the same convention: "Has anyone stopped to consider that the best way to balance the federal budget is not by taxing people into the poorhouse and it's not by cutting spending to the bone, but rather it's by all of us simply trying to live up to the Ten Commandments and the Golden Rule?" Two days later, the president was posing for pictures with Afghan leaders when he was asked a question about arms-control progress at Geneva. "You know I'm not answering questions at press conferences . . . ," he responded, confirming what some reporters have suspected all along.