John, Anne, or Peter (is/are) going to do it.
Giving your mother flowers or buying a card (is a great idea / are great ideas) for Mother's day gift.
I am aware of the rule saying that two singular subjects connected by or or nor require a singular verb. My American friends say that "is" does not sound natural in the first sentence and "are" sounds valid in the second one (as well as "is"). I am a bit confused
What grammar rule have you used? Is the alternative option grammatically correct or incorrect? If it was incorrect, why?