Characterizing all models in infinite cardinalities

Fix a cardinal κ. We can ask the question what kind of a logic L is needed to characterize all models of cardinality κ (in a finite vocabulary) up to isomorphism by their L-theories. In other words: for which logics L it is true that if any models A and B satisfy the same L-theory then they are isomorphic.
It is always possible to characterize models of cardinality κ by their Lκ+,κ+- theories, but we are interested in finding a "small" logic L, i.e. the sentences of L are hereditarily smaller than κ. For any cardinal κ it is independent of ZFC whether any such small definable logic L exists. If it exists it can be second order logic for κ = ω and fourth order logic or certain infinitary second order logic L2κ,ω for uncountable κ. All models of cardinality κ can always be characterized by their theories in a small logic with generalized quantifiers, but the logic may be not definable in the language of set theory.