gh auth status should distinguish transport failures from invalid credentials #12891

Summary

gh auth status currently reports that a token is invalid when the underlying failure is actually at the network/transport layer. In restricted environments this sends users toward re-authentication even though their credentials may be fine.

Example Use Case

A concrete case is running gh auth status inside a sandboxed or otherwise restricted environment where DNS or outbound network access is blocked. In that situation the human-readable output says the token is invalid, but the actual failure is a transport error such as lookup api.github.com: no such host. Running the same command outside the restricted environment succeeds without re-authentication.

This can also apply in CI, containers, devboxes, or corporate network setups where GitHub is temporarily unreachable or DNS/proxy configuration is broken.

Reproduction

1. Run gh auth status in an environment with blocked DNS or blocked outbound access to GitHub.
2. Observe that the human-readable output says the token is invalid and suggests gh auth login or gh auth logout.
3. Run gh auth status --json hosts or GH_DEBUG=api gh auth status.
4. Observe that the real error is transport-related, for example lookup api.github.com: no such host.
5. Run the same command outside the restricted environment and see authentication succeed.

Current Behavior

• Human-readable output reports an authentication problem.
• Machine-readable/debug output can already show the real transport error.

Expected Behavior

• Human-readable output should distinguish invalid credentials from transport/network verification failures.
• Re-authentication guidance should only be shown for actual invalid-credential failures.
• If maintainers prefer a minimal change, JSON output can remain backward compatible while the human-readable messaging is corrected.

Additional Context

I already implemented a draft fix in PR #12878, which keeps the JSON shape stable and only changes the human-readable classification/message path.