DALLAS — Fight or flight?

Those are, in essence, the two very different approaches that the activist community is taking with Exxon Mobil these days.

One group is made up of investors who want to hold on to their stakes in the company and prod the and gas giant, through shareholder resolutions and by applying steady pressure, to adopt new policies that reflect the realities of .

The other major group is campaigning to make Exxon Mobil a pariah and is urging big institutional shareholders to dump all their fossil-fuel stocks.

Both groups showed up in force on Wednesday at Exxon’s annual shareholder meeting here. Outside, several dozen protesters waved signs reading “ARREST EXXON” and “KEEP IT IN THE GROUND,” and chanted slogans like “Exxon lies, oceans rise!” and “Windmills, not toxic spills!”

Nearby, a 13-foot-wide block of ice cut into the shape of the hashtag — the shorthand used to call attention to the company’s early research into climate change — melted in the morning’s mugginess.

Inside the Morton H. Meyerson Symphony Center, where Exxon Mobil held its meeting, activist shareholders sat among the company’s loyalists. Their ranks included corporate crusaders like Sister Patricia Daly, a Dominican Sister of Caldwell, N.J., who is executive director of the . That group is a member of the , which represents more than 300 institutional investors with more than $100 billion in assets.

“Our voice has been very important,” said Sister Daly, on corporate policies that affect sustainability and human rights.

Shareholders placed 11 before the board, several of them related to climate change and corporate governance.

The climate resolutions were voted down on Wednesday, but a “proxy access” resolution that called for the company to give large shareholders a bigger say over corporate governance passed with nearly 62 percent of the votes. Two other proposals related to corporate governance also passed.

A proposal calling for the company to report on how climate change regulations would affect its business received 38.2 percent of the vote. Advocates said that was the biggest showing ever for a climate resolution at an Exxon meeting and a sign of increasing support for such measures over time.

Exxon Mobil’s chief executive, Rex Tillerson, addressed climate change as part of his overall remarks at the annual meeting, citing the company’s extensive research over the decades and its work with the .

“We believe that addressing the risk of climate change is a global issue” that will require the cooperation of governments, business and individuals, he said, and he repeated the company’s support of a carbon tax.

Mr. Tillerson also predicted that oil and gas would long be an important part of the company’s mix, because “abundant energy supplies are vital for modern life.”

Demand for oil and gas will remain high, he said, though “stringent energy policies will increase the cost of carbon dioxide emissions over time.” That would further depress the fortunes of coal, he added, but would bolster the use of natural gas, nuclear power and renewable energy.

The company, he said, had worked tirelessly “to understand and reduce the risk of climate change.”

The activist shareholders, of course, disagree, and are impatient to see Exxon Mobil change its ways. But is one approach more effective than the other?

Some have a big-tent philosophy, seeing the value in efforts to both work with and assail the company.

“It’s wonderful that the divestment people have built up so much momentum among people who are not shareholders anyway and don’t plan to be,” said Sister Daly, the lead filer of a shareholder resolution calling for Exxon Mobil to (3.6 degrees Fahrenheit).

But others take a more one-sided view. , a leader of the divestment movement and co-founder of the group , said in an interview that the time for shareholder engagement had passed.

By his count, there have been 62 shareholder resolutions asking Exxon Mobil to act on climate change, and none have passed, as a majority of shareholders have sided with management to vote them down.

“It might finally be time for shareholders of conscience to simply break ties with this company,” he said. “Fool me 62 times, shame on me — and in fairly large quantities.”

The campaign to persuade institutions to dump their fossil-fuel holdings can point to many successes. Globally, the movement to drop at least some fossil fuel stocks has grown to 500 organizations that have sold off an amount estimated by 350.org at .

Yet on college campuses, where the movement began roughly four years ago, divestment drives appear to be meeting some resistance. This spring’s crop of protests on campus did not produce major victories at premier schools.

In April, Stanford University , stating that its advisory panel on responsible investing “concluded that it could not evaluate whether the social injury caused by the fossil fuel industry outweighs the social benefit it provides, and therefore did not recommend divestment.” (That panel did recommend divesting from extraction, but Stanford didn’t have such investments.)

At New York University last month, 18 students occupied a campus elevator for 33 hours. The elevator led to the offices of top administrators; the school threatened the students with suspension — which could have made it impossible for them to enter on-campus housing or to take their final exams — and contacted their parents.

Olivia Rich, one of the protesters, said parents began calling them, saying, “You’ve got to get out of the elevator!” Although the school did not agree to sell any investments, N.Y.U. did agree to some of the students’ demands, including a meeting with the school’s investment committee.

“I would say it was a win,” Ms. Rich said.

Schools that decide against divestment often say that they support action against climate change in their classrooms and labs but that their endowment, as , “is a resource, not an instrument to impel social or political change.”

The divestment activists say their push makes economic as well as moral sense. They argue that oil and other energy companies will inevitably face a reckoning, just like the coal industry, now that most nations have pledged to take the steps necessary to slow global warming after last year’s climate talks in Paris.

But the simple act of divesting may carry its own costs. A new report commissioned by the Independent Petroleum Association of America cited in the so-called frictional costs of divestment — the transactions required to get the schools out of the mutual funds that make up the bulk of most schools’ endowments and move to fossil-free funds or handpicked portfolios.

Those costs, wrote Professor Hendrik Bessembinder of Arizona State University, the author of the report, would “range between approximately 2 and 12 percent of the endowment’s value, which, for a typical large university endowment, would translate to a decline in value of between $1.4 billion and $7.4 billion” over a 20-year period.

But those organizing the protests say that focusing on the losses of individual funds misses the point. As with the campus divestment protests about South African apartheid in the 1980s, much of the activism is focused on changing the national — and global — conversation.

“Students are leading the way and shifting the narrative on climate,” said Sara Blazevic, the campus strategy coordinator of the Fossil Fuel Divestment Student Network. “We are winning the fight over the story of the climate crisis.”

Beyond the campus, other organizations have declined to sever ties with the industry. This month, the United Methodist Church decided against divestment, despite one of its prominent members, Mr. McKibben.

And in a much-debated decision, the American Geophysical Union declined to drop Exxon as a sponsor at events such as a student breakfast during the group’s fall meeting.

The group’s president, Margaret Leinen, that the $35,000 breakfast “does not constitute a threat to the group’s reputation.” But hundreds of members and other scientists have disagreed, pressing for a complete break with the company.

The Methodist decision, like many others, is at least in part about staying in the game as the more effective way to lobby for change.

“If we divested tomorrow, we’re not going to put Exxon Mobil out of business,” said Thomas P. DiNapoli, New York State comptroller. “And one of the things we will lose is the ability to move them in a different direction as a shareholder.”

The divestment fight, he argued, glosses over shared responsibility for climate change.

“Every organization, every business, has a fossil fuel footprint,” he said. “We have a real challenge out there that goes beyond whether you have Exxon Mobil stock.”

That responsibility, he added, is personal as well.

“It’s great to say you’re for divestment,” he said. “But if you get in your car and fill the gas tank on the way to the rally, it undercuts your argument.”