The answer to your specific question, "Is it the body dubbing color you use or the hackle color that defines each?", if we're talking about a somewhat 'basic adult' caddis pattern, then the answer is - the body color (dubbed, yarn, whatever) - defines the name. Of course there is a sh*tload of patterns that are more intricate and they get 'non-basic' names. Then there's another sh*tload that are developed (or have minor alterations to existing patterns) and they are promoted under the name of the 'developer'!
But back to your point - a Tan, Green, Grey, Black Caddis, etc. are caddis' with that color as the body.

Have fun and tight wraps.

Allan