Is a Karp–Levin reduction a Levin reduction?
A Karp reduction from a language $A$ to a language $B$ is a function $f$ such that $x \in A$ iff $f(x) \in B$. This is the common meaning of the term.

David Richerby mentions in a comment below that Levin reductions are a more elaborate type of reductions: see this question.

In the present instance, Karp–Levin reduction seems to mean the same as Levin reduction. Indeed, since the common term for the usual notion of many-one reduction is Karp reduction, adding Levin's name might signify something different. The way the term is used in the paper should make it clear which meaning is being used. The paper might also include a definition of this somewhat non-standard term, and I suggest you include such a definition in future papers you may write.