By RICK LYMAN

HOUSTON — First, Judge Sandra Watts was stopped while trying to vote because the name on her photo ID, the same one she had used for voter registration and identification for 52 years, did not exactly match her name on the official voter rolls.

A few days later, state Senator Wendy Davis, a Democrat who became a national celebrity after her filibuster over a new abortion law, had the same problem in early voting. So did her likely Republican opponent in next year’s governor’s race, Attorney General Greg Abbott.

They were all able to vote after signing affidavits attesting that they were who they claimed to be. But not Jim Wright, a former speaker of the House in Washington, whose expired driver’s license meant he could not vote until he went home and dug a certified copy of his birth certificate out of a box.

On Tuesday, Texas unveiled its tough new voter ID law, the only state to do so this year, and the rollout was sometimes rocky. But interviews with opponents and supporters of the new law, which required voters for the first time to produce a state-approved form of photo identification to vote, suggest that in many parts of the state, the law’s first day went better than critics had expected.

There was a relatively limited number of cases during early voting in which voters with improper IDs were required to submit provisional ballots, which will be counted only if the people come back with a valid ID within six days. Officials said that statewide, 2,354 provisional ballots were cast this election, which is about 0.2 percent of voters. In the last off-year election, in 2011, there were 738, or 0.1 percent of the ballots cast that year.

Officials also said that there was little traffic at the offices set up by the state to provide free voter ID documents for those without another approved form of identification. By Election Day, only 121 voter identification documents had been issued statewide.

But the law’s opponents and some county elections officials warned against judging its effectiveness on the basis of this year’s relatively minor statewide election. The real tests, they say, will come next year, when Texas will elect a new governor, and in 2016’s presidential race.

“This kind of off-year election draws the more experienced voters who turn up all the time,” Dana DeBeauvoir, the Democratic clerk of Travis County, said. “This is not the more casual voter who turns out for a presidential election.”

Critics also raised concerns that the law might disproportionately affect women, because of name changes from marriage or divorce. But there were no hard figures on how many women were actually affected versus men — and there were many instances of men who were stopped for the same reason.

The nonpartisan League of Women Voters of Texas, which opposed the new law, said that it was concerned more about voters who do not have the proper documentation at all, and might stay away from the polls altogether as a result.

“We have always felt there was anywhere from 500,000 to 800,000 voters who would not be able to present the proper identification,” Linda Krefting, the group’s president, said. “The concern we have is that all this flap in the news may have discouraged people from turning out at the polls.”

Republican leaders who had pushed for the new law said that for an off-year election, which included several mayor’s races and statewide constitutional proposals, voting was robust. Election officials said Wednesday that 1,144,844 people had voted, compared with 690,052 in 2011, the previous such off-year election.

“This was our first statewide election with a photo ID requirement in place, and it was smooth, secure and successful,” Secretary of State John Steen said Wednesday in a statement.

Voter ID laws and other statutes that cut back on early voting or make it more difficult to register have proliferated in states dominated by Republicans since the party’s wave of governor and statehouse victories in 2010.

Proponents say they are needed to curtail voter fraud. Opponents point out that such fraud is extremely rare and say the laws actually target groups that have proven less likely to have the state-mandated identification: the poor, students, African-Americans and Hispanics, all of whom tend to vote more Democratic.

Under the new Texas law, the list of acceptable identification includes a driver’s license, a passport, a military ID and a concealed gun permit, but not a student photo ID. Voters who showed up at the polls with no acceptable IDs were allowed to cast provisional ballots. Voters whose names were “significantly similar” on their IDs and the official voter rolls could sign an affidavit, which involved checking a box next to their name, then were allowed to vote normally.

More than 30 states have passed laws, with varying degrees of strictness, that either require or request that voters produce some form of identification. But there have been recent signs of a backlash.

The Brennan Center for Justice said 10 states passed 13 bills to expand voting compared with eight states that passed nine restrictive bills this year.