Expose current auth method (subscription vs API key) in status line payload

Feature request

Currently the status line JSON payload (session_id, cwd, model, workspace, version, cost, context_window, output_style, exceeds_200k_tokens) has no field indicating whether the active session is authenticated via a Pro/Max subscription login or via ANTHROPIC_API_KEY (pay-as-you-go).

Why this matters

Per the documented auth precedence, ANTHROPIC_API_KEY silently overrides an active subscription login when both are present. Today the only way to check which credential is actually in use is running /status manually. For users who deliberately set ANTHROPIC_API_KEY temporarily (e.g., prefixing a single command to burn through a subscription usage spike via pay-as-you-go), there's no persistent, glanceable indicator of which billing mode is active — making it easy to lose track and unintentionally rack up API charges in a later session, or conversely think you're on API billing when you're actually still on the subscription.

Request

Add a field to the statusLine payload (e.g., auth_mode: "subscription" | "api_key", or similar) so a custom status line script can display it, the same way cost and model are already surfaced.

Related: #24679 (requesting account/email in the payload) was closed as a duplicate — this request is specifically about auth method, not account identity, so filing separately.