The original story was also factually incorrect, as the website had failed to note Vine is not a BBC News presenter.
Questions were raised in the 9am BBC News editorial meeting chaired by director of news Fran Unsworth, according to BuzzFeed News.
The BBC website then had to check with the BBC press office whether the BBC could confirm the names of the BBC presenters who had taken cuts, and it was decided that the story must be credited to the BBC News media editor rather than the BBC itself.
A source said: "This hasn't been handled very well."
At lunchtime, the BBC press office put out a statement that read:"We are very grateful to Huw Edwards, Nicky Campbell, John Humphrys, Jon Sopel, Nick Robinson and Jeremy Vine, who have agreed that their pay will now be reduced.
"These are great journalists and presenters, who have a real connection with the audience. We are proud to have them working at the BBC.
"The final details of some of these changes are still being discussed, and there are further conversations that the BBC will have with others in due course."
Humphrys has agreed to cut his salary for the Today programme by NZ$290,000-$385,000 (£150,000-£200,000).
Vine has already taken a cut, saying the decision "wasn't a problem" as the BBC needs to tackle gender pay inequality.
Campbell agreed to a cut in order to bring his pay close to that of his Radio 5 Live breakfast co-presenter Rachel Burden.
He said pay negotiations had been "all very civilised and collegiate".