Prefer env var auth over default codex auth (#1861)

## Summary
- Prioritize provider-specific API keys over default Codex auth when
building requests
- Add test to ensure provider env var auth overrides default auth

## Testing
- `just fmt`
- `just fix` *(fails: `let` expressions in this position are unstable)*
- `cargo test --all-features` *(fails: `let` expressions in this
position are unstable)*

------
https://chatgpt.com/codex/tasks/task_i_68926a104f7483208f2c8fd36763e0e3

codex-rs/core/tests/client.rs

@@ -458,7 +458,7 @@ async fn azure_overrides_assign_properties_used_for_responses_url() {
         request_max_retries: None,
         stream_max_retries: None,
         stream_idle_timeout_ms: None,
-        requires_auth: false,
+        requires_openai_auth: false,
     };
 
     // Init session
@@ -481,6 +481,86 @@ async fn azure_overrides_assign_properties_used_for_responses_url() {
     wait_for_event(&codex, |ev| matches!(ev, EventMsg::TaskComplete(_))).await;
 }
 
+#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
+async fn env_var_overrides_loaded_auth() {
+    #![allow(clippy::unwrap_used)]
+
+    let existing_env_var_with_random_value = if cfg!(windows) { "USERNAME" } else { "USER" };
+
+    // Mock server
+    let server = MockServer::start().await;
+
+    // First request – must NOT include `previous_response_id`.
+    let first = ResponseTemplate::new(200)
+        .insert_header("content-type", "text/event-stream")
+        .set_body_raw(sse_completed("resp1"), "text/event-stream");
+
+    // Expect POST to /openai/responses with api-version query param
+    Mock::given(method("POST"))
+        .and(path("/openai/responses"))
+        .and(query_param("api-version", "2025-04-01-preview"))
+        .and(header_regex("Custom-Header", "Value"))
+        .and(header_regex(
+            "Authorization",
+            format!(
+                "Bearer {}",
+                std::env::var(existing_env_var_with_random_value).unwrap()
+            )
+            .as_str(),
+        ))
+        .respond_with(first)
+        .expect(1)
+        .mount(&server)
+        .await;
+
+    let provider = ModelProviderInfo {
+        name: "custom".to_string(),
+        base_url: Some(format!("{}/openai", server.uri())),
+        // Reuse the existing environment variable to avoid using unsafe code
+        env_key: Some(existing_env_var_with_random_value.to_string()),
+        query_params: Some(std::collections::HashMap::from([(
+            "api-version".to_string(),
+            "2025-04-01-preview".to_string(),
+        )])),
+        env_key_instructions: None,
+        wire_api: WireApi::Responses,
+        http_headers: Some(std::collections::HashMap::from([(
+            "Custom-Header".to_string(),
+            "Value".to_string(),
+        )])),
+        env_http_headers: None,
+        request_max_retries: None,
+        stream_max_retries: None,
+        stream_idle_timeout_ms: None,
+        requires_openai_auth: false,
+    };
+
+    // Init session
+    let codex_home = TempDir::new().unwrap();
+    let mut config = load_default_config_for_test(&codex_home);
+    config.model_provider = provider;
+
+    let ctrl_c = std::sync::Arc::new(tokio::sync::Notify::new());
+    let CodexSpawnOk { codex, .. } = Codex::spawn(
+        config,
+        Some(auth_from_token("Default Access Token".to_string())),
+        ctrl_c.clone(),
+    )
+    .await
+    .unwrap();
+
+    codex
+        .submit(Op::UserInput {
+            items: vec![InputItem::Text {
+                text: "hello".into(),
+            }],
+        })
+        .await
+        .unwrap();
+
+    wait_for_event(&codex, |ev| matches!(ev, EventMsg::TaskComplete(_))).await;
+}
+
 fn auth_from_token(id_token: String) -> CodexAuth {
     CodexAuth::new(
         None,