[Feature Request] Distinguish between API key and OAuth subscription billing in startup banner

Bug Description
▎ The Claude Code startup banner labels the auth path as "Claude API", which suggests developer-API/credit billing. However, the same wording appears when the session is
▎ authenticated via CLAUDE_CODE_OAUTH_TOKEN (Claude.ai Pro/Max subscription), where usage bills against the subscription plan rather than API credits.
▎
▎ This is confusing — users can't tell from the banner whether they're consuming API credits or subscription quota. Suggest distinguishing the two (e.g. "Claude API" vs "Claude
▎ subscription / Pro / Max") in the welcome message and/or /status.
▎
▎ Repro: start claude with only CLAUDE_CODE_OAUTH_TOKEN set (no ANTHROPIC_API_KEY); banner still reads "Claude API".

Environment Info

• Platform: darwin
• Terminal: ghostty
• Version: 2.1.117
• Feedback ID: a0c2b879-1085-460e-acfe-5ffde6bfba23

Errors

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