Grams...bullet weight, or powder weight?

Cartridge boxes routinely give the bullet weight on the label. By and large, the propellant isn't identified, either the type or weight, and putting the weight of an unknown propellant on the label would be useless anyway, since the energy content of each is different. The quasi-exception is shotshells, which sometimes still list a "dram equivalent" that approximates the energy content (but not type or actual weight) of the propellant by saying how it compares to so many drams of black powder. And don't apologize over the gram/grain thing. I'm looking at my VihtaVuori manual, which uses both.

It can be either, but on the side of a box of bullets, it's referring to bullet weight.

Grains is a unit of measurement for the weight of the bullet itself. Apparantly, 1 grain is equal to 64.79 mg.

Do you mean grains? On a box of bullets this would usually refer to the projectile weight. But it can also refer to powder when refering to reloading. It just depends on the context. For factory loaded ammunition the grains listed on the box refer to bullet weight. I have a box of .223s in front of me that say 55gr, meaning that the the projectiles weigh 55 grains.