Autotransformer vs Inductive Divider
An autotransformer and an inductive divider are different animals. There are places where one would work, and not the other. You would use each when it's the right tool for the job.

An autotransformer is wound with both windings on a common core. An inductive divider uses two individual inductors, with no shared flux.

The most significant difference in behaviour is that an autotransformer has a low impedance output, in the ideal case, zero. This means it can be used to step down (or to step up) an AC supply, and the load can draw significant power. Its winding inductances are large, not well specified, and in the ideal case infinite. It's not something you would resonate with capacitors and expect to get reproducible results.

An inductive divider looks like an autotransformer plus an output impedance of the two inductors in parallel, just like a resistive divider looks like a lower voltage with an output impedance of the two resistors in parallel.

This means you would rarely use an inductive divider to power a load, its output would sag, just like a resistive divider. It's often found in tuned circuits though, with its well defined inductances the correct values to resonate with capacitors.