One of my coworkers passed along some information that clears up a fine distinction:
An eyelet, strictly speaking, is a single-piece metal ring, reinforcing a hole in material like fabric or paper. You’ll often find these in office-supply stores…
A grommet is bigger than an eyelet, and is two pieces (a base and a washer). It’s another way of reinforcing a hole in material with metal: the grommet base goes through a hole, the washer is placed over it and the two are affixed to each other. A specialized kind of grommet is the 
bead core, used to make a silver lining in your bead, giving a 
“Pandora-style” effect.
Rivets look like eyelets from the back, but their tops are solid, with no hole, and are rounded. Prior to use, a rivet is a smooth cylindrical shaft; one end has a head on it.
A 
decorivet is a jewelry finding made by Vintaj. It’s a decorative metal object (butterfly, compass, blue-jeans style rivet, etc.) with a pointed rivet shaft that can be pressed through or folded around other surfaces.
And some off-subject trivia: the (usually plastic) tip at the end of a shoelace (or bolo cord) is an aglet.
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