What's the difference between logic (in a narrow sense, i.e. a logical system such as ZOL, FOL, etc.) and type system?
I will sketch my understanding of this -- please correct if I err. Under propositions-as-types, they coincide. In propositions-as-some-types, a logic(al system) is a proper subset of the type system. In logic-enriched type theory (LTT), they are distinct components. "Untyped" always means "single-typed" (as in untyped $λ$-calculus, which has only functions), and describes the object level of the formalism (i.e. its objects belong to a single type); together with its meta level there are more types (e.g. the syntactic operators $λ$, . in $λ$-calculus). I'll finish with some examples of the possible combinations:
typed logic : FOL
typed nonlogic : a data type system
untyped logic : untyped lambda-calculus*
untyped nonlogic : a teapot
* not a logic but can encode one