I think the Passho, Coba, etc. Berries are the berries that Serebii is referring to. You don't normally find those berries lying around on the ground, and were originally the harder-to-get berries.
This is a good question though, and this question may be on the fence, in terms of a right answer. "Rare," in terms of berries, could mean anything, because odds of getting certain berries from Mom are random.