Neither of those rules are strictly true, and they're very often not true at all. Tire rolling resistance is mostly a product of sidewall construction and rubber compound. For example, compare the popular WTB Byway with a slick center tread and the Continental Race King MTB tire with knobs -- the MTB tire with knobs is about 50% faster.
Also, when comparing two tires of identical construction and differing only in width, the wider one will have less rolling resistance at the same pressure.