In my opinion, the Wikipedia article on complete theory (mathematical logic) contains a mistake, or at least is ambiguous and misleading.
The first paragraph is fine, it gives the correct definition of complete theory in mathematical logic.
The second paragraph is problematic. It says:
This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness. This theorem states that recursively axiomatizable first-order theories that are rich enough to allow general mathematical reasoning to be formulated cannot be complete.
I agree with the first two sentences. The last sentence is wrong (or at least misleading) because it attributes the content of Gödel's first incompleteness theorem to Gödel's completeness theorem, in this way not only the information is mistaken but it also confuses two different meanings of completeness.
As far as I know, in mathematical logic there are two different senses for "completeness", one for theories (for instance, Presburger arithmetic is complete, Peano arithmetic is not) and one for logics (for instance, first-order logic is complete, second-order logic is not). Gödel's completeness theorem is about completeness of first-order logic (completeness of a logic), Gödel's first incompleteness theorem is about (in)completeness of Peano arithmetic (completeness of a theory). Clearly, there is no contradiction between the two Gödel's theorems.
Do you agree with me? Please, can someone fix the error in the Wikipedia article?