Add model provider info to /status if non-default
Enterprises are running Codex and migrating between proxied / API key
auth and SIWC. If you accidentally run Codex with `OPENAI_BASE_URL=...`,
which is surprisingly easy to do, we don't tend to surface this anywhere
and it may lead to breakage. One suggestion was to include this
information in `/status`:
<img width="477" height="157" alt="Screenshot 2026-01-09 at 15 45 34"
src="https://github.com/user-attachments/assets/630ce68f-c856-4a2b-a004-7df2fbe5de93"
/>