I doubt Giuliani was misquoted, or he would have trumpeted it from the rooftops. So your complaints about the Post don't come into it. I think Giuliani can be believed here.

There is a video of Giuliani saying these things, so if you would like to see it, just do a search. In fact, it is a (possibly the) critical argument being made in the legal case this evening.

Giuliani literally said on Fox News. Watch the actual video, don't just complain about media bias. The evidence is right there in the linked article.

“I'll tell you the whole history of it,” Giuliani responded eagerly. “So when [Trump] first announced it, he said, 'Muslim ban.' He called me up. He said, 'Put a commission together. Show me the right way to do it legally.' "

Better source for you?