meteorite - Are micrometeorites cosmic dust?

Micrometeorites are just tinier versions of regular meteorites. As such, they can have the same composition as any other meteor class. There's nothing that we know of that says they have a chemistry more like one kind of dust source, although this is controlled entirely by observational bias. Certain types of particles are more likely to make it to our planetary neighborhood.

We know that interstellar dust grains are found in meteorites, as are all kinds of other grains. Most (or many) meteorites we have contain grains from all over our cosmic neighborhood, including those formed long before our Earth, and they're conglomerated together in the meteorite matrix. Micrometeorites are smaller and will have less types of grains and likely more weathering due to their higher surface area to volume ratio.

To answer your question: are micrometeorites cosmic dust?

Some probably contain microscopic bits of cosmic dust, but some won't, and those that do aren't necessarily entirely made of cosmic dust.