After a seven-day tour of some of the key developing countries of the world, President Carter headed for home yesterday apparently satisfied that he has dramatized the United States' concern for the people of the Third World.

On his flight here from Lagos, Nigeria, the president told reporters aboard Air Force One that he considered his trip to Venezuela, Brazil and Nigeria to be "much better than we had anticipated in every way."

But even as Carter and other American officials were expressing satisfaction with trip, it was clear that the Third World leaders who met with Carter will not be satisfied unless the symbolism of the journey is followed up by concrete U.S. action.

Carter arrived here yesterday morning in the most stiflingheat and humidity the American party has encountered in the last week.

Thousands of people jammed Monrovia's airport, clashing with police and soldiers at times as the crowd tried to get closer to the site of the welcoming ceremony.

The announcer on Liberian radio exclaimed that "the crowd had gone wild."

As Carter rode the 40 miles from the airport of Monrovia, the capital, people from villages along the route cheered and waved flowers and palm fronds.

The crowd in Monrovia, estimated at 100,000 about half the city's population, was by far the largest Carter has encountered on this trip.

During his visit here of only a few hours - "a four-hour lunch break," one Liberian paper called it - Carter had a working lunch with President William Tolber and met with 250 American volunteers at a Peace Corps center.

Despite its long and close ties with the United States, Liberia had never had a state visit by a U.S. president, although Franklin D. Roosevelt stopped here briefly in 1943 on his return from the Casablanca conference in Morocco.

Leaving Nigeria earlier yesterday, U.S. officials were cheered by what they saw as an indication from the country's leader, Lt. Gen. Olusegun Obasanjo, of sympathy for the American concern over the presence of the Cuban troops in Ethiopia.

Obasanjo said Sunday that Africans have suffered for too long from "fratricidal and often futile wars, in many cases with the encouragement by powers from ideological camps outside the continent who are seeking ideological, economic and strategic spheres of influence."

"It is Africa's desire to settle her own disputes in her own way, if necessary under the auspices of the Organization of African Unity," he added.

Carter has stressed his concern over the presence of Cuban troops in Africa repeatedly on this trip, but until Sunday night there had been no public response from Obasanjo.

Meanwhile, U.S. officials disclosed the first details growing out of Carter's announcement in Lagos Sunday that the United States is moving to convene a peace conference involving all parties in the dispute over black majority rule in Rhodesia.

Secretary of State Cyrus Vance said the United States and Britain issued invitations yesterday to two meetings, the first a preliminary session and the second the planned conference of all parties.

Vance said leaders of the Patriotic Front, which is fighting the Rhodesian government of Ian Smith, as well as the foreign ministers of the so-called "front line" black-ruled states that surround Rhodesia have agreed to attend both meetings.

Vance said the United States has not yet heard from Smith or the domestic black leaders who earlier reached an "internal settlement" for majority rule with the Smith government.

Earlier, Smith and the Rhodesian-based black leaders who entered into the agreement with him said they were not interested in such a meeting and called on the United States and Britain to support the internal settlement.

Vance said, "We certainly hope that they will come because if one wants to find a way to stop the fighting then you have to get all the parties together."

Carter and Obasanjo were clearly in accord on the Rhodesian question. Carter's reaffirmation of his support for the Anglo-American plan for black majority rule, rather than the internal settlement, was a prime objective of Nigerian and other black African leaders.

But there was much less agreement on the question of South Africa, where the United States is resisting black African demands for a total economic boycott of the minority white government.

Obasanjo suggested Sunday that the economic future of the United and other Western powers in Africa might be determined by their handling of the South African question.

He said he hoped that United States is beginning to realize that "put together, black Africa as a whole offers wider economic possibilities as an alternative choice than South Africa alone, and that without a peaceful change of the police of apartheid now, any investment in South Africa is a risky and insecure investment."

The president, in his airbone interview, acknowledged that differences remain between him and Obasanjo on the question of sanctions against South Africa. But he also warned the South Africans that their relations with the United States are reaching a crucial stage.

If South Africa rejects "a reasonable proposal" for the U.N.-supervised elections in Namibia (Southwest Africa) he said, it could "precipitate more serious differences between us and South Africa." He did not elaborate.

The United States, along with most of the world, has demanded that South Africa turn over rule of Namibia to a government elected in U.N.-supervised elections open to the Southwest African Peoples Organization, a guerrilla group that has fought for Namibia's independence.

South Africa has agreed to grant independence to Namibia, but is trying to ensure the election of a government friendly to South Africa.

At the arrival ceremony in Monrovia, Carter sought to sum up the purpose of his 14,500-mile journey to Venezuela, Brazil, Nigeria and Liberia.

"In coming to Liberia, I am reaffirming a friendship that is very old, but I am also drawing to a close a series of visits that reflect a world that is new," he said. "Less than three decades from now, four-fifths of all the world's people will live in Africa, Asia and Latin America - in the sorts of developing nations that I have visited.

"Only three decades ago, many nations of these continents were largely colonies of foreign powers," Carter continued. "Their rise to independence means a world in which we must treat each other as equals, and one of the purposes of these trips has been to demonstrate the genuine respect my nations feels for its partners around the world and our opposition to the continuation or reestablishment of colonialism in any form whatsoever."

That was the message that Carter sought to convey at every stop, dramatizing the shift in U.S. foreign policy under his administration to place more emphasis on the developing nations of the Third World.

As such an exercise in international symbolism, the trip, as it neared its conclusion, appeared to have been a success. Moreover, there were indications that the president got along well with the Third World leaders he met, even if he did not resolve differences on specific issues.

There is no question that differences remain.

Officials of Venezuela and Nigeria made it clear to Carter that they will support an oil price increase, which would put more pressure on the dollar.

There were also no agreement reached with Brazilian officials over the issues of human right and nuclear nonproliferation, or with the Nigerians on South Africa.

Overriding all of these specific issues, however, was the sense, strongly conveyed to Carter by each of the foreign leaders, of Third World impatience with what they consider empty promises by the developing nations.

By becoming the first American president to visit Latin America since John F. Kennedy and the first to make an official visit to a black nation, Carter more than ever committed the United States to a growing alignment with the Third World.

The real test, especially in the eyes of the leaders with whom Carter met, will come in the months ahead when the administration will be expected to follow through on the president's rhetoric.