President Carter and Ronald Reagan headed onto the campaign trail yesterday to begin their final week's run for the presidency, with advisers for each camp claiming that their candidate had significantly helped his prospects for victory with Tuesday night's debate.

As the president campaigned in the Northeast and his Republican challenger swung south into Texas, the biggest confrontation of the day occurred in Washington, in the battle of the polltakers. Featured were Reagan pollster Richard Wirthlin and Carter's Patrick Caddell, each serving up contradictory sets of statistics and analyses.

The events began with Wirthlin calling a news conference to say: "Our polling clearly indicates Gov. Reagan won the debate." He cited two polling efforts, one a nationwide survey by a Michigan firm, the other a survey in which 200 Seattle residents rated the candidates' handling of each question in the Tuesday debate.

The latter survey showed Reagan besting Carter on four questions asked, Carter winning two and two even, Wirthlin said.

Back at the Carter headquarters, Caddell convened a news conference of his own to say this was not so. He said he had just spoken with the Seattle pollster, R. D. Percy, and that Percy had authorized him to say that the actual results were Carter four and Reagan four -- and that Carter had rated considerably better on the final, closing statement of the debate.

Yet Caddell was not claiming a total Carter victory in Cleveland. "On the question of who won," he said, "I think it is basically a wash, with perhaps Reagan having a slight advantage."

But he went on to say that determining the victor is not the same as determining which candidate benefited the most from the debate performance. And on that, Caddell maintained, Carter helped his cause significantly with statements that will appeal to the undecided voters -- and especially those who are traditionally Democratic voters and who are being lured back to the Carter fold.

As the Reagan and Carter pollsters were offering their assessments, another national survey indicated that the event had been essentially a draw in the minds of America's voters.

The CBS-New York Times poll surveyed 1,259 people before the debate, and then reached 1,019 of them again afterward. Of this sample, which turned out to consist of slightly more Reagan than Carter supporters, 83 percent watched at least some of the debate. Forty-four percent said they thought Reagan had won, and 36 said Carter.

The margin of error is plus or minus 4 percentage points. And more significant, perhaps, was that 90 percent of those interviewed said that the debate did not cause them to change their mind about their vote.

Late in the day, Caddell publicly lamented that it is an "unfortunate role" into which pollsters have forced themselves. "We're all running around playing games -- racing around playing 'Who won?'. . . The question is who was impacted the most, how were they impacted, and how will it affect the electoral vote, if at all?"

On the Reagan campaign plane, Leadership '80, there were optimistic expressions from the candidate and his staff that Reagan had accomplished his major purpose in his 90-minute debate with Carter. That purpose, his aides said, was to show that Reagan is "presidential" -- a feat he accomplished by performing on an equal footing with the president.

"I think anyone watching the debate would conclude that Reagan was a reasonable man and that he had command of the facts and the issues -- perhaps surprisingly so," said James Baker, a Reagan adviser who was the GOP nominee's chief negotiator in the debate.

Reagan's message on the campaign trail throughout this final week will be largely the same one he has been trying to deliver for the last two weeks -- that Carter's record of performance, especially on economic issues, is one of failure.

"What we have to do is what we did in the debate last night," he said. That is, "demonstrate that Ronald Reagan is a reasonable alternative to a failed presidency."

Meanwhile, on the Carter campaign plane, Air Force One, presidential press secretary Jody Powell offered this assessment of the debate results and the campaign prospects for the final week:

"I think it was the turning point. I think the movement was generally in the president's direction anyway, before the debate, and I think the results of the debate . . . probably will give a little impetus to it."

One Carter adviser added, "We're not going to take many of his voters and he is not going to take many of ours. The battle was over the undecideds, and I think Carter hit them on every issue."

Reagan's chief polltaker, Wirthlin, produced the results of figures from the firm of Market Opinion Research, a firm operated by Republican pollster Robert Teeter. The firm surveyed 500 people after the debate. Forty-five percent said Reagan did a better job, and 34 percent said Carter did better. The margin of error is plus or minus 4.5 percentage points.

Among independent and undecided voters, Reagan was the winner, according to this survey. But it is impossible to draw meaningful analysis from these figures, because there was no data provided on whether these were independents and undecideds who had been leaning toward Reagan originally.

These figures offered by Wirthlin showed 35 percent saying they would be more likely to vote for Reagan because of the debate, and 30 percent saying they would be more likely to vote for Carter -- a difference that is not significant due to the margin of error.

Caddell said that one of the major Carter strategies was to keep the debate focused on the issues that are Carter's strength -- foreign policy, war and peace and social issues such as the Equal Rights Amendment and Social Security and Medicare -- and away from the economy. This was accomplished he said. And on this, Caddell and his Reagan counterpart seemed to agree.

Wirthlin said that from watching the debate he was convinced that Reagan had won. But then he read the transcript of the debate the next day, and he said: "Thank God it was on television."

So it was yesterday that advisers from both camps participated in that classic campaign ritual of declaring debate victory, win, lose or draw.

In the debate audience in Cleveland Tuesday, shortly before the event was to start, Republican vice presidential nominee George Bush was asked, jokingly, by a reporter if he would mind giving his post-debate analysis in advance, to avoid any last-minute rush.

"Not at all," Bush laughed, as he stood in the Reagan VIP section (front row, right) five minutes before the debate began. "Gov. Reagan was truly presidential. He demonstrated leadership and great depth in his handling of the issues."

The reporter, continuing this predebate repartee, asked Bush whether Reagan had really meant to liberate Bulgaria in this debate (a reference to President Ford's 1976 debate gaffe about Poland's not being dominated by the Soviets), Bush joked right back:

"Well, you have to understand that we are talking about the spirit of the people of Bulgaria being liberated, not the actual government of Bulgaria itself. You see, it's all very clear now -- and that's why Gov. Reagan has won this debate so convincingly."

After the debate, Bush offered his assessment for real, as he stood in front of a group of reporters and cameras and lights.

"Clearly Gov. Reagan emerged as presidential," Bush said, in words that rang familiar. "He projected a certain hope and leadership on the issues, and he resisted getting mired down in answering outrageous charges. Gov. Reagan was just way ahead, way ahead."