This must be an example of a U.S./British usage difference, because in the United States it would definitely be "on the exam." I know this for sure, because I went to a high school that practice the honor system. Classes voted whether or not to be on the honor system. If your class was on the honor system, you had to write at the top of an exam, "On my honor, I know of no cheating on this test" and sign it. Quick Googling seems to confirm the U.S./British hypothesis: "The most elaborate ways to cheat in your exams..." --UK source. "The University operates a zero tolerance policy in relation to cheating in examinations." --UK source. "Cheating in exams is fairly common in the Indian state of Bihar..."--UK source "Promotions frozen for about 200 NYPD sergeants amid probe into alleged cheating on exam..." --U.S. source. "Doctors cheated on exams."--U.S. source. "Goldman traders cheated on internal exam."--U.S. source. "Prevent Cheating on Exams"--Canadian source. I also see "cheating at exams," and "cheating during exams."

"On the exam" is just wrong. It is not controversial at all. She cheated on your husband, but you cheated in the exam.

I would use "in the exam" to talk about the event, and "on the exam" to talk about the paper.

I can't find a consensus for this question, but I will say that, "I did well ON the test/exam, " is how I would say it. I came across some comments that make it seems like British English speakers prefer, "I did well In the test/exam, " which might explain why the Cambridge Dictionary words it that way.