Are micrometeorites just dust grains at high relative speed, or is there some difference between them and cosmic or interstellar or solar nebula dust? Dust as in the clouds which obscure parts of the Milky Way in our night sky.
It might seem like the way to figure this out would be to test the composition of micrometeorites and see what objects this matches up with. This, however, is not a viable technique, as Genge et al. (2008) explains:
In interplanetary space dust particles are inherently transported sunwards by P-R light drag (Dohnanyi 1967) and once captured by the Earth their deceleration occurs at high altitudes allowing fragile micrometeoroids to survive atmospheric entry (Love and Brownlee 1991). The parent bodies of MMs, therefore, are likely to differ from those of meteorites.
It is, however, possible to analyze micrometeorites and group them in various categories, and determine the parent bodies. The two main types of parent bodies are thought to be comets and asteroids. Distinguishing between the micrometeorites of these two groups, however, can be difficult. Many large micrometeorites are from main-belt asteroids (see Genge & Grady (2002)). However, Nesvorny et al. (2010) suggests that many micrometeorites (and large amounts of cosmic dust in general) are from comets.
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.