"Natural" reductions vs "Polynomial-time many-one" reductions (Karp Reductions)
Natural reductions are a special case of Karp reductions, with an extra condition stating that the output's length depends only on the input's length (and not structure). This concept can be applied to many forms of many to one reductions, so I guess you could talk about natural logspace reductions.

Obviously a karp reduction can be non-natural (otherwise the definition would be of no use). Think about a reduction from clique to SAT, I can make it non natural by first checking whether the graph is planar, and in case the answer is yes output some fixed non satisfiable formula (suppose the parameter $k$ is greater than $4$). This breaks naturality, since now the output formula's length depends on the planarity of the input graph.

Finding an example of a "natural" non-natural reduction (even if a natural one is known) is harder, but I guess that if you go over enough reductions you will find one.