Refactor auth providers to mutate request headers (#17866)

## Summary
- Move auth header construction into the
`AuthProvider::add_auth_headers` contract.
- Inline `CoreAuthProvider` header mutation in its provider impl and
remove the shared header-map helper.
- Update HTTP, websocket, file upload, sideband websocket, and test auth
callsites to use the provider method.
- Add direct coverage for `CoreAuthProvider` auth header mutation.

## Testing
- `just fmt`
- `cargo test -p codex-api`
- `cargo test -p codex-core
client::tests::auth_request_telemetry_context_tracks_attached_auth_and_retry_phase`
- `cargo test -p codex-core` failed on unrelated/reproducible
`tools::handlers::multi_agents::tests::multi_agent_v2_followup_task_interrupts_busy_child_without_losing_message`

---------

Co-authored-by: Celia Chen <celia@openai.com>

codex-rs/codex-api/src/endpoint/responses_websocket.rs

@@ -1,5 +1,4 @@
 use crate::auth::AuthProvider;
-use crate::auth::add_auth_headers_to_header_map;
 use crate::common::ResponseEvent;
 use crate::common::ResponseStream;
 use crate::common::ResponsesWsRequest;
@@ -310,7 +309,7 @@ impl<A: AuthProvider> ResponsesWebsocketClient<A> {
 
         let mut headers =
             merge_request_headers(&self.provider.headers, extra_headers, default_headers);
-        add_auth_headers_to_header_map(&self.auth, &mut headers);
+        self.auth.add_auth_headers(&mut headers);
 
         let (stream, server_reasoning_included, models_etag, server_model) =
             connect_websocket(ws_url, headers, turn_state.clone()).await?;