Add additional call types to `api_callers`

In `api_callers`, we have base classes for three API request methods: GET, POST, and DELETE.

There are additional methods, like PUT and PATCH.

Follow the idioms established in the `api_callers` module and implement classes for each of the remaining request types.

Start in the api_callers module by reading the existing GET, POST, and DELETE base classes and following their established idioms. Add corresponding classes for PUT and PATCH, then verify that each remaining request type is represented consistently with the existing classes.

Written by the indexing model from the issue text.