Mutual definability does not imply definitional equivalence, a simple example
Hajnal Andréka, Judit X. Madarász, Istvàn Németi · Mathematical logic quarterly · 2005
We give two theories, Th1 and Th2, which are explicitly definable over each other (i.e. the relation symbols of one theory are explicitly definable in the other, and vice versa), but are not definitionally equivalent. The languages of the two theories are disjoint. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)