The application cited claims from the dossier that Mr. Page, while on a trip to Moscow in July 2016, had met with two senior Russian representatives and discussed matters like lifting sanctions imposed on Russia for its intervention in Ukraine and a purported file of compromising information about Mr. Trump that the Russian government had. (Mr. Page has denied those allegations, although he later contradicted his claims that he had not met any Russian government officials on that trip.)

Republicans portrayed the Steele dossier — which also contained salacious claims about Mr. Trump apparently not included in the wiretap application — as dubious, and blasted the F.B.I. for using material from it while not telling the court that the Democratic National Committee and the Hillary Clinton campaign had funded the research.

But Democrats noted that the application also contained evidence against Mr. Page unrelated to the dossier, and an unredacted portion of the application discussed efforts by Russian agents in 2013 to recruit Americans as assets. It has previously been reported that Mr. Page was one of their targets, although any discussion of Mr. Page’s interactions with them in the application is still censored.

Democrats argued in February that the F.B.I. had told the court that the research’s sponsor had the political motive of wanting to discredit Mr. Trump’s campaign. They argued that it was normal not to specifically name Americans and American organizations in such materials. The released documents show that portion of the filings, which the previously released Democratic memo had quoted.

The application shows that the F.B.I. told the court it believed that the person who hired Mr. Steele was looking for dirt to discredit Mr. Trump. But it added that based on Mr. Steele’s previous reporting history with the F.B.I., in which he had “provided reliable information,” the bureau believed his information cited in the application “to be credible.”

The applications largely avoided using names; renewal materials noted that they would continue to refer to “Candidate #1” by that description, for example, even though he “is now the president.”

The renewal applications from 2017 told the court in boldface print that the F.B.I. had severed its relationship with Mr. Steele because he had shared some of his claims with a news organization in October 2016, contrary to the F.B.I.’s “admonishment” to speak only to law enforcement officials about the matter. But they said the bureau continued to assess his prior reporting as “reliable.”