Lillian Carter was going to campaign against a Republican congressman from Missouri. Now she is having second thoughts: the Republican voted with the president on the water bill veto.

Rep. James J. Florio (D-N.J.), who also voted with the president, thinks he has reason to believe the White House will support some public works projects he has wanted for his district.

A California congressman says he was told his state might lose a defense contract if he voted wrong.

For a year and a half now, President Carter has been the student, and House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) the No. 1 tutor as Carter has learned how to deal with a frequently balky Democratic Congress.

In the shootout over the water bill yesterday, pupil and tutor were on opposite sides, and the pupil president won. Moreover, he did it at least in part in traditional style, with an old fashioned mixture of carrot and stick, treats and promises.

The White House campaign ended in victory yesterday when the House fell 53 votes short of the two-thirds margin required to pass the appropriation bill over the veto.

Although the floor speeches and the press releases that accompanied yesterday's vote would suggest that every member decided the question with only the public interest in mind, members of Congress were trading stories all day about personal and political interests that swayed individual decisions.

Such stories accompany every big struggle on the Hill, of course, but members on both sides of yesterday's vote agreed that the arm-twisting on the public works veto was almost unparalled in the two years since Carter came to Washington.

The president had almost every member of his Cabinet working the telephones this week. "I got a call from the secretary of defense," said Rep. James F. Lloyd (D-Calif.), a member of the Armed Services Committee. "I thought he was going to talk about [aircraft] carriers. No, he says, he just thought he'd ask me to vote for the public works veto."

Lobbyists from nearly every government agency were dispatched to the House side of the Capitol yesterday to seek last-minute conversions. There were so many jammed together outside the doors of the House chamber when the vote started at 1:45 p.m. that the corridor looked like the Tokyo subway at rush hour.

The president needed all the firepower he could muster, because he was opposed by the entire Democratic leadership of the House. The leadership had a potent weapon - a threat that members who supported the veto might lose out in later public works bills.

"They told me that the Public Works Committee was keeping a list of everybody who votes to sustain [the veto]," said a junior member from the Midwest. "It was a kind of way of saying I'll never get another project for my district."

Carter countered that kind of pressure by promising several members that he would support their pet projects in the next Congress.

"When the president called me, I told him I was worried about the veto, because there are some good projects in the bill," said Florio, the New Jersey Democrat. "And he told me that they would go through in the next bill without any trouble."

The White House showed it could be accomodating in other ways as well.

When Rep. E. Thomas Coleman, a freshman Republican from Missouri, got a call Wednesday from Frank Moore, the chief White House lobbyist, he told Moore he was inclined to back the president.

"But I told him it was a little aggravating to do it," Coleman said, "because Chip Carter was in my district this week campaigning for my opponent. And Miss Lillian is going out there next week.

"And Frank said, maybe the Carters could rethink that trip."

The president focused much of his personal effort on first-and-second-term members, who are less accustomed than their senior colleagues to the pork-barrel ethic that no member of Congress questions another's project.

That emphasis turned out to be precisely right, because the junior members voted most heavily to support the veto.

Carter also paid unusual attention to Republicans. Several junior GOP members received personal calls from the president, and yesterday morning 30 republicans were invited to a working breakfast at the White House, where Carter told them that he was deeply appreciative of Republican support on such earlier bills as the Civil Service law changes and executive branch reorganization.

"That made an impression," said Rep. James A. S. Leach (R-Iowa). "It was good to hear him say that because we have helped him on something and we've never gotten much credit for it."