Your understanding is correct, and in your case, you will have to provide a criminal record certificate from the U.S. unless there is a good reason of why you should not, or cannot, provide it.
What are the requirements for applicants' criminal record certificate for the U.K. tier 1 investor visa?
I noticed that investors of the U.K. tier 1 investor visa program need to provide a criminal record certificate for any country that he or she has stayed in "for a total of 12 months or more over the past 10 years." During the past 10 years I have been visiting the U.S. as a visitor almost every year, each time for about two months. If my understanding is correct, my accumulative stay in the U.S. during the past 10 years has exceeded 12 months and I would need to provide a criminal record certificate from the U.S. But every time I visited I was a temporary visa holder and I am not sure such a thing as "criminal record for visitor" even exists in the U.S. system. Am I interpreting this requirement incorrectly?