`core`: Feign Implementation Details exposed to clients

The use of the Feign library/framework is an implementation detail and shouldn't concern the consumers of the library.

However, every API call throws runtime exceptions all of which extend Feign's. That they are Runtime does abstract away many concerns. However, the APIs should throw declared exceptions (Runtime or not) so that consumers can properly handle error states, without relying on inspecting the underlying error details.

Start by inspecting the core API entry points and the runtime exceptions currently exposed by their Feign-backed calls. Define the public exception boundary and identify how callers should handle declared error states; done means API consumers no longer need to depend on Feign exception types.

Written by the indexing model from the issue text.