Is a rivulet really a river, only smaller?

A rivulet is a flow of liquid. You might see rivulets coming from your sink as you brush your teeth, streaming from the gutters on your house after a rain, or leaking out of a crime scene victim during TV prime time.

From Henry Fielding's Joseph Andrews:

Adams continued his subject till they came to one of the beautifullest spots of ground in the universe. It was a kind of natural amphitheatre, formed by the winding of a small rivulet, which was planted with thick woods, and the trees rose gradually above each other by the natural ascent of the ground they stood on . . .

And as Sinclair Lewis wrote in Main Street:

Rivulets were hurrying in each alley; a calling robin appeared by magic on the crab-apple tree in the Howlands' yard.

In George Eliot's Silas Marner:

. . . these too belonged to the past, from which his life had shrunk away, like a rivulet that has sunk far down from the grassy fringe of its old breadth into a little shivering thread, that cuts a groove for itself in the barren sand.