I know, it's a bit confusing. That's why I wrote:

I'm trying to generate all multisets (bags) of a specific total "weight" (let's call it w), where each element comes from a given list (of numbers, in this case), and each list element may have multiplicity 0..w in each multiset.

and:

The order in which the multisets itself are generated isn't important to me either, BTW. I've only listed them in order for the sake of readability.

I'm sure there's standard terms for these, too, terms that I simply don't know.