Is every conservative extension an extension by definitions?

Question

Suppose we have a theory T and a theory T' with corresponding languages L and L', where T' is a conservative extension of T. Must it be the case that T' proves that every relation symbol or constant symbol or function symbol that is in L' but not in L, has a "definition" in terms of only L? I apologize if my question is naive or trivial.

Answer 1

Let $T$ be any complete first-order theory in the language $L$ with infinite models. Let $L'$ be any proper extension of $L$. Let $M\models T$, and let $M'$ be an expansion of $L$ to $L'$ which interprets at least one of the new symbols in $L'$ as a relation or function on $M$ which is not $L$-definable. Let $T' = \text{Th}(M')$.

Then $T'\supseteq T$, and $T'$ is not an extension by definitions. But $T$ is already complete, so $T'$ cannot entail any $L$-sentences which are not already entailed by $T$, so $T'$ is a conservative extension of $T$.