Francisco Javier Remes Sánchez was puzzled as he watched President Trump sign an executive order last week renaming the Gulf of Mexico to the Gulf of America as part of his pledge to honor “American greatness.”

“That man talks a lot and we have no choice but to defend Mexico,” said Mr. Remes Sánchez, 52, who manages a 15,000-member fishing association in Tamaulipas state in northeastern Mexico. He has fished in the gulf for 20 years and estimated that he spends 2,000 hours a year on its waters.

“He’s changing the name of a cultural and natural heritage of Mexico since the 16th century, when the United States hadn’t even been formed,” he added.

To be clear: Mr. Trump’s order renaming the world’s largest gulf only changed the name in the United States, where he has authority, not internationally. He asked the Secretary of the Interior to remove all mentions of the Gulf of Mexico in the government’s official geographic database and ensure that “all federal references,” including maps, contracts and other documents, reflected the new name.

On Friday, the Interior Department announced the switch.

But still, across Mexico and Cuba, the other countries with maritime boundaries in the gulf, Mr. Trump’s move was met with a combination of bewilderment, indignation, indifference and, at times, laughter.

“For us and for the whole world, it is still the Gulf of Mexico,” President Claudia Sheinbaum of Mexico said, briefly chuckling when addressing the topic last week.

For some, renaming the gulf reminded them of other global disagreements over place names. For example, the body of water south of Iran has long been a source of tension, with Iran, like much of the world, calling it the Persian Gulf, while Saudi Arabia and some other Arab states prefer the Arabian Gulf.

The Rio Grande, the river along the southern edge of Texas that serves as the national border between the United States and Mexico, is called that on the northern side. But in Mexico it is known as the Río Bravo.

The gulf has had many names, from the Gulf of Florida to the Gulf of Cortés, but there is evidence of the Gulf of Mexico name sticking as early as 1552, used in print by a Spanish historian, said Samuel Truett, a professor at the University of New Mexico who specializes in U.S. and Mexican history.

Even though the name came nearly 300 years before the country of Mexico was founded, its origins are from the Aztecs, who built a city on which Mexico City was later erected. He noted that while people from the United States typically use “America” to mean their country, the term long predates the nation and originally meant something much broader. To many Latin Americans, it still does.

“It’s this U.S. conceit of taking something that was applied to the hemisphere, really, and claiming it only for one nation,” he said.

When Mr. Trump first suggested he would alter the gulf’s name earlier this month, Ms. Sheinbaum showed a world map from 1607 that labeled North America as Mexican America and identified the Gulf of Mexico as such — 169 years before the United States was founded.

“Why don’t we call it Mexican America? It sounds pretty, no?” she joked then.

Since Mr. Trump signed the order, a few other Mexican officials have defended the gulf’s name.

Rocío Nahle, the governor of Veracruz state, which has over 450 miles of gulf coastline, wrote the morning after the signing: “For 500 years, it is and will continue to be our rich and great ‘Gulf of Mexico.’” After rattling off the reasons it was important to her state, from commerce to petroleum to fishing, she wrote, “It is not a decree, it is reality!!”

Even the country’s national tourism ministry chimed in, saying in a social media post promoting visits to the body of water: “Long live the Gulf of Mexico! The beauty of our Mexico is marvelous, before the eyes of the world and as it has been called since 1607 on world maps.”

(So far, the Cuban government hasn’t said anything on the subject.)

In Tampico, a port city in Tamaulipas state, José Antonio Zapata Hinojosa, 45, an economics and political science professor, as well as a local amateur historian, said that people there were unconcerned with Mr. Trump’s decision. He said he imagined that everyone — even tourists — will continue saying the Gulf of Mexico.

“It’s like when they change the name of a street or a stadium, the original name still stays,” he said.

In Cuba, Edel Pérez, 54, runs a hotel with his family in Santa Lucía, a town on the northwestern side of the island, from which he can see the gulf. He said he has been fishing for grouper, snapper and more in those waters all of his life.

“I don’t understand how a person on a whim wants to change the name,” he said. “Our part will continue being the Gulf of Mexico.”

Mr. Pérez said he was astonished when he first heard the news but admitted that the Cubans he knew would likely shrug their shoulders. “The people here don’t worry about that stuff.”

He said it was curious, though, that Mr. Trump, seeking to glorify the United States, chose the name “America,” since “we are all Americans.”

Oddly enough, the most popular and successful Mexican soccer team is called Club América, which has become fodder for memes on social media in Mexico.

Practically, though, the gulf’s name doesn’t matter, said Capt. Paul Foran, a maritime consultant based in Florida who as a ship captain navigated through the Gulf of Mexico countless times.

While it might be a lot of work to change all of the U.S. government documents and programs, Captain Foran said, mariners only worry about using the correct navigation charts and conveying the proper coordinates and speed to nearby ships over the radio.

“The guy on the other end listening to me say ‘Gulf of America,’ he’s going to look at his chart and he doesn’t care what it’s called,” he said. “All he will care about is, ‘OK, I see that guy, I know where I am. I’m in the Gulf of Mexico and he’s calling it the Gulf of America. Who cares? Just don’t run into me.’”

The gulf’s name could change again in four years when Mr. Trump’s term is over, said Mr. Truett, the history professor in New Mexico.

But if Mr. Trump’s “America first” logic were to be applied elsewhere in the United States, wouldn’t the state of New Mexico also be vulnerable to a name change?

No, Mr. Truett said while laughing, he wasn’t worried that his state’s name would be switched to “New America” anytime soon.