I have every faith that most lawyers keep records. I have more faith than normal that Cohen, knowing who Trump was (even if he was still working for him on purpose), made and kept more records than usual.
Yeah, I have to admit, at some point Team Trump is going to question Cohen's trustworthiness and man do I wish I was the judge for just that one issue. The idea of Trump saying "you cannot trust the word of the lawyer I used for decades, he admitted he lied and committed crimes on my behalf" is just too much and needs to be strangled to death in its own irony. But if I'm right and Cohen has records that back his side of the story, and Trump doesn't, it's an attempt that won't get past the judge. Might work on the jury, tho.