“It really spoke to the power of Apple controlling the operating system,” said Brian Wieser, president of business intelligence at GroupM, an advertising industry firm. “Facebook isn’t in control of its own destiny.”

At Facebook, Apple’s privacy moves were viewed as hypocritical, said three current and former Facebook employees. Apple has long had a lucrative arrangement with Google to plug Google’s data-hungry search engine into Apple products, for instance. Facebook executives also noted that Apple was entrenched in China, where the government surveils its citizens.

Privately, Mr. Zuckerberg told his lieutenants that Facebook “needed to inflict pain” upon Apple and Mr. Cook, said a person with knowledge of the discussions. The Wall Street Journal previously reported Mr. Zuckerberg’s comment.

Behind the scenes, that work had already begun. In 2017, Facebook had expanded its work with Definers Public Affairs, a Washington firm that specialized in opposition research against its clients’ political foes. Definers employees distributed research about Apple’s compromises in China to reporters, and a website affiliated with Definers published articles criticizing Mr. Cook, according to documents and former Definers employees.

Definers also began an “astroturfing” campaign to draft Mr. Cook as a 2020 presidential candidate, presumably to put him in President Trump’s cross hairs, The New York Times reported in 2018. A website, “Draft Tim Cook 2020,” featured a lofty quote from the chief executive and a model campaign platform for him. Data behind the website linked it to Definers.

(Definers’ work against Apple was also funded by Qualcomm, another Apple rival, according to a Definers employee. Facebook fired Definers after The Times reported on its activity.)

Apple and Facebook have also started competing in other areas, including messaging, mobile gaming and “mixed-reality” headsets, which are essentially eyeglasses that mix digital images into a person’s view of the world.