Per my comment in tokio-rs#1313, this improves discoverability of how `State`
works with nested/merged `Router`s.

I made a few other minor documentation fixes while I was reading:

* Corrected the docs for `Router::nest` to say that it nests another
  `Router` rather than a `Service`. There is a separate function
  (`Router::nest_service`) for the latter.
* Changed one of the headings in nest.md to use more idiomatic English.
* Changed awkward phrasing under the "Sharing state with handlers"
  heading of the root documentation page.
* Removed a trailing period from one of three list items for
  consistency.

Co-authored-by: David Pedersen <david.pdrsn@gmail.com>

Co-authored-by: David Pedersen <david.pdrsn@gmail.com>