From deciding when President Reagan should be told about an aerial battle with Libyan planes to briefing reporters on the president's behalf, Edwin Meese III this week displayed the unique public role he plays, one that has won him the nickname "President Meese."

Every president since World War II has had one or two close advisers whose influence, combined in varying degrees with control of access to the Oval Office, has given them great power.

None of Meese's predecessors has been so public, however, in asserting that power.Few chose to be so often seen and photographed in the spotlight, instead seeking a lower profile and letting the president, other elected officials and Cabinet department heads play the more public parts.

Meese began the week in his role as chief spokesman, briefing reporters about a National Security Council meeting. He ended it counseling Reagan not to answer reporters' questions during a brief photographic session in the president's hotel suite.

In between, Meese decided not to awaken Reagan with news of the dogfight near Libya, and accompanied the president to the carrier Uss Constellation, where he sat down in the VIP row with the president, wearing an identical windbreaker and nearly identical baseball-style cap.

Reagan's cap was labeled in gold braid "Commander in Chief." The counselor's was labeled "Ed Meese." It was the most recent of several occasions when Meese has been placed in the presidential row rather than remaining in the background, as was the pattern of earlier White House advisers.

The carrier visit came a day after Meese delayed for six hours telling Reagan about the dogfight, in which two Navy pilots shot down two Libyan pilots over the Gulf of Sidra.

Reagan said Meese had done the right thing. "Yes, 4:30 in the morning California time is as early as I want to be awakened," the president said with Meese standing at his side nodding approval.

Later, the president said there was a very good defense for that decision. "If our planes were shot down, yes, they'd wake me up right away. If the other fellow's were shot down, why wake me up?" he said to laughter and applause from an audience of California Republicans.

Meese, it was explained, had not thought Reagan should be awakened, because the dogfight was long over and there was no presidential decision to be made.

At his Monday account of the National Security Council meeting, Meese spoke in generalities, and was at times glib and evasive.

Asked if the president had made any decisions, he replied: "None at all." Asked why, he replied: "Because he wants to make them later."

In a Tuesday interview with Ken Bode of NBC, Meese was even less forthcoming. Meese said the budget could be cut without crimping plans for additional defense spending and without injuring important social programs. A number of unnecessary boards and commissions could be eliminated painlessly, he said.

He was hard-pressed to name any of them, finally citing one that he thought was a good candidate for elimination if it had not been abolished already in the first round of budget cuts.

Like a lawyer counseling a client, Meese broke in with a chuckle Friday when reporters tried to question Reagan about the MX missile during a photo opportunity.

"Mr. President, you're not obliged to answer any questions," Meese said to laughter from the president and two visiting members of Congress. A suggestion by Judy Woodruff of NBC that the president might want to answer questions was lost in the merriment.

A minute later, a reporter asked Reagan why the administration's decision on how to base MX missiles was slipping behind schedule.

Meese interrupted to deny the question's premise. Things are right on schedule, he said, with a decision due in three or four weeks.

Meese's role was particularly visible this week in Los Angeles, where he was the only senior Reagan aide in attendance throughout the week, and therefore stood out more clearly than he might have during a working week in the White House.

Whether here or in Washington, however, Meese's public prominence fills a void created by Reagan's decision to discuss issues only rarely. Although the president makes frequent short speeches in and out of the White House, a Republican fund-raising party Wednesday night marked the first time he has taken questions from an audience.

The president has held three White House news conferences and one extended informal question-and-answer session with reporters at his ranch.

There are also the almost daily photo sessions, lasting only a couple of minutes, during which reporters ask the president questions that Reagan aides often try to squash before the president can answer.

Despite White House aides' apparent dislike of exposing Reagan to these questions, the photo sessions are useful to the White House because they often provide the only live pictures of Reagan for that evening's TV news.

Meese also is thrust into the spotlight because of the sometimes puzzling manner in which presidential decisions are timed.

Decisions frequently are announced several days or weeks after the meetings at which Reagan has participated in extensive discussions on an issue.

For example, Reagan came to last Monday's National Security Council meeting after 10 days on his ranch, and announced he had decided to resume shipments of military aircraft to Israel. That decision presumably could have been made public much earlier, because no one claimed that Reagan had been consulting on the issue while at his ranch.

The same pattern was followed in lifting the Soviet grain embargo and in other cases.

This system leaves Meese, who is a Cabinet member and controls the flow of foreign and domestic policy information, in the apparent position of determining when many presidential decisions are to be announced.

It also creates a tension between reporters and Meese. The reporters want to hear from Reagan how and why a decision was reached. Often Meese has offered an explanation and wonders why reporters are putting to Reagan questions that Meese has already answered.

Although Meese and other senior White House aides bristle at the suggestion that Reagan has delegated so much authority that he is not making all the decisions himself and is not keeping up with all developments, the image sometimes slips.

When the Justice Department released a study on violent crime this week, White House deputy press secretary Larry Speakes was asked what Reagan thought of it. Speakes consulted his briefing notes and replied that Reagan thought the report was a comprehensive job done in a remarkably short time.

He conceded, however, that Reagan had not had time to read the report. The president's high opinion of it, reporters were told, was based on what Meese and others had told him.