A (first-order) theory is $\kappa$-categorical if all its models of cardinality $\kappa$ are isomorphic. Now is there a similar notion which doesn't require isomorphism, but only elementary equivalence? (Let's say "elementary categorical".) Some questions about:
1) Are there any theories with this property (for some $\kappa$ for which they have a model of cardinality $\kappa$) which are not $\kappa$-categorical? And are there any which are not complete?
2) Are there any theories which satisfy the property for every $\kappa$, and are not $\kappa$ categorical for any $\kappa$? And are there any which are not complete?
Sorry for the lack of preciseness in the question, but being a somehow "unheard" problem for me I didn't really know how to state it.
Thank you in advance.