admin: Rename `cache` package to `caching`

This package both:

A) Stutters because it contains a `Cache` struct
B) and is confusing to new developers

We should rename this package to `caching` similar to how we have a `logging` package that containers a `Logger` struct.

Start by locating the `cache` package and searching the repository for references to its package name and `Cache` struct. Rename the package to `caching`, update all references, and run the repository's test suite to confirm that the rename is complete.

Written by the indexing model from the issue text.