Natural weapon damage is weapon damage, right?

Natural weapons are not weapons for the purposes of rules/text, but natural weapon damage is weapon damage, right?

— TormentedByChristmas (@TrmntdByGnomes) October 27, 2018

There are weapon attacks, and there are spell attacks.

There are also damage types: bludgeoning, piercing, lightning, and so forth.

Weapon damage is not a damage type in D&D. #DnD https://t.co/hYrS2H5oux

— Jeremy Crawford (@JeremyECrawford) October 27, 2018