NEW YORK — It wasn’t until after a decade in the fold, after his family pleaded with him, after the FBI raided his office, apartment and hotel room, Michael Cohen testified Tuesday, that he finally decided to turn on Donald Trump.

That decision led to a 2018 guilty plea to federal charges involving a payment to the porn actor Stormy Daniels to bury her story of an alleged sexual encounter with Trump and to other, unrelated crimes. And it’s that insider knowledge of shady deals that pushed Manhattan prosecutors to make Cohen the star witness in their case against Trump about that same payment, which they say was an illegal effort to influence the 2016 presidential election. Under questioning this week, Cohen has described the nuts-and-bolts of how the scheme worked.