Use `URL`s instead of strings for config values

Using strings to hold user-defined URL configurations is a bit unwieldy. They should instead be `URL`s to allow some nicer processing on them (or maybe even some external URL-parsing library can be used, as `URL` is finicky in its own ways).

Start by tracing how user-defined URL configuration values are represented and processed throughout the project. Compare the behavior and limitations of TypeScript URLs with the current string handling, including whether an external URL-parsing library is needed; done means the configuration approach is agreed and applied consistently.

Written by the indexing model from the issue text.