context_server: Abstract server transport (#24528)

This PR abstracts the communication layer for context servers, laying
the groundwork for supporting multiple transport mechanisms and taking
one step towards enabling remote servers.

Key changes centre around creating a new `Transport` trait with methods
for sending and receiving messages. I've implemented this trait for the
existing stdio-based communication, which is now encapsulated in a
`StdioTransport` struct. The `Client` struct has been refactored to use
this new `Transport` trait instead of directly managing stdin and
stdout.

The next steps will involve implementing an SSE + HTTP transport and
defining alternative context server settings for remote servers.

Release Notes:

- N/A

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Co-authored-by: Marshall Bowers <git@maxdeviant.com>