You are right about the confusion deriving from the name 'kataifi.' It is surely used for both the dough as well as the concoctions made out of it. Perhaps the dough (which is indeed extruded as you guessed) should be called 'fillo kataifi' but this is rarely the case and for good reason as 'fillo' means 'sheet' in Greek and kataifi does not match this description at all. Concerning the finished product, it is indeed baklavas filling wrapped in kataifi dough and dipped in essentially the same syrup as baklavas.
Contrary to kataifi, the baklavas dough is called 'fillo kroustas' in Greek with the latter word meaning nothing that I or DH can trace. However, it does serve to create a clear distinction between the names of raw material and finished product in the case of baklavas. You said it correctly, go figure!
The proof of the pudding is in the eating!