That was always my understanding of the term, too -- until I got into the subject of European royal genealogies. I think COUSIN is basically an umbrella term referring to any relative, other than a parent, sibling, aunt, uncle (or great-aunt, great-uncle, etc.) with whom one shares a common direct ancestor.

It could mean first cousin, first cousin, once removed, second cousin, etc. However, when not specified, COUSIN seems to usually mean first cousin.