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/tui/src/lib.rs

@@ -287,7 +287,7 @@ fn restore() {
 
 #[allow(clippy::unwrap_used)]
 fn should_show_login_screen(config: &Config) -> bool {
-    if config.model_provider.requires_auth {
+    if config.model_provider.requires_openai_auth {
         // Reading the OpenAI API key is an async operation because it may need
         // to refresh the token. Block on it.
         let codex_home = config.codex_home.clone();