[Bug]: Cursor API-key auth is treated as logged out

Before submitting

• I searched existing issues and did not find a duplicate.
• I included enough detail to reproduce or investigate the problem.

Area

apps/server

Steps to reproduce

Cursor CLI documents API keys as a first-class auth method: https://cursor.com/docs/cli/reference/authentication

1. Install Cursor CLI. Do not run agent login.
2. Export a valid CURSOR_API_KEY (or pass --api-key). See the same auth doc and https://cursor.com/docs/cli/reference/parameters
3. Confirm the CLI itself works: cursor-agent models lists models.
4. Confirm cursor-agent about --format json returns "userEmail": null (no saved login).
5. Enable the Cursor provider in T3 Code on that same machine / environment.

Expected behavior

T3 treats the official API-key session as authenticated, discovers models, and can start a Cursor thread. Same shape as Grok's XAI_API_KEY path.

Actual behavior

1. Provider status is unauthenticated: Cursor Agent is not authenticated. Run \agent login` and try again.`
2. After that check is satisfied, ACP model discovery fails. T3 always sends ACP authenticate with cursor_login, and the CLI tries to open a browser even though CURSOR_API_KEY is already enough for session/new and cursor/list_available_models.

T3's Cursor probe only looks at agent about userEmail (apps/server/src/provider/Layers/CursorProvider.ts) and hardcodes authMethodId: "cursor_login" (apps/server/src/provider/acp/CursorAcpSupport.ts). It never reads CURSOR_API_KEY.

Impact

Major degradation or frequent failure

Headless / CI / API-key-only hosts cannot use Cursor in T3 even when the CLI is healthy.

Version or commit

main @ d23b181da

Environment

macOS, Cursor CLI 2026.08.11, T3 Code server driving cursor-agent

Logs or stack traces

cursor-agent about --format json
{
  "cliVersion": "2026.08.11-e8db854",
  "userEmail": null,
  "subscriptionTier": null
}

ACP authenticate / cursor_login:
{"error":{"code":-32602,"message":"Invalid params","data":{"message":"Failed to open browser for login. Please visit: https://cursor.com/loginDeepControl?..."}}}

session/new and cursor/list_available_models succeed with the same CURSOR_API_KEY if authenticate is skipped.

Workaround

Run agent login so about has an email. That is the interactive path, not the documented API-key path.

Start with apps/server/src/provider/Layers/CursorProvider.ts and apps/server/src/provider/acp/CursorAcpSupport.ts, then reproduce the API-key-only flow with CURSOR_API_KEY and cursor-agent about --format json. Trace the provider authentication check and ACP authenticate request. Done means T3 recognizes the documented API-key session, discovers models, and can start a Cursor thread without opening a browser.