Startup screen shows 'Claude API' instead of subscription plan when authenticated via OAuth

Bug description

When launching Claude Code with an active Pro Max subscription authenticated via OAuth, the startup screen displays:

Opus 4.6 with high effort · Claude API

This is misleading — it suggests the session is using the pay-per-token API rather than the subscription plan.

Expected behavior

The startup screen should display the actual subscription plan (e.g., Pro Max or Max) instead of Claude API when authenticated via OAuth/first-party.

Actual behavior

Claude API is shown regardless of the authentication method.

Verification

Running claude auth status returns the correct state:

{
  "loggedIn": true,
  "authMethod": "oauth_token",
  "apiProvider": "firstParty"
}

So the auth is correct — only the display label is wrong.

Environment

• Claude Code v2.1.86
• macOS (Darwin 25.4.0)
• Plan: Pro Max (X20)

Impact

Cosmetic but confusing — users may think they're being billed on the API when they're actually on their subscription, or vice versa. Can cause unnecessary support tickets.