To be clear, I am a criminal defense attorney who handles appeals, so, I'm not suffering from a "misconception." Either you are or you misunderstood something. This is going to be vague, since being specific can take a long time in law. It is only about attorney-client privilege:
1. A client telling you about a crime he committed is covered by the attorney-client privilege.
General rationale: The defense attorney's job is to zealously defend the charged client. To do this, communications between client and attorney as to what is charged need to be unrestrained. If the client did it, he needs to tell the lawyer exactly what happened so that the lawyer can craft the best defense. That is impossible if, the moment the client admits committing the crime, the admission is suddenly not "covered by the attorney-client privilege." Clients could not communicate anywhere near fully with their attorneys, and their attorneys would almost certainly do a much poorer job as they would be operating on bad information. This is a simplification.
2. There are narrow exceptions.
a. The attorney may not aid the client in committing future crimes. Anything passing between attorney and client in service of commission of future crimes is not privileged. This is a simplification.
b. As I said, the client accuses the attorney of things in certain forums and the attorney must violate privilege in order to defend himself, ie, malpractice lawsuit ("you didn't tell me blah blah blah, so I did this, and I got screwed!"). This is a simplification.
There are others.
But " attorney client privilege does not cover crimes" is broad to the point of complete falsehood. If Trump tells Guiliani about past crimes, that's privileged. IF Guiliani helps Trump commit future crimes via legal advice, that isn't privileged. Same, if Trump throws Giuiliani under the bus as to past crimes in such a way that fit the narrow exception where the client has put the attorney in a position where revealing privileged info is sufficiently necessary to the defense of the attorney.
This all comes down to what Guiliani thinks his "insurance" is and whether Trump puts him in one of the narrow situations where he may ethically reveal that information.