[codex] fix Claude AI Gateway key messaging

Summary

This updates the template copy and key-selection flow so Claude is presented as an AI Gateway-backed agent instead of requiring an Anthropic key.

What changed

• pin the Claude and Codex API key check route to aigateway
• update the API keys dialog to say Claude and Codex use AI Gateway
• hide the Anthropic key entry unless a user already has a legacy saved Anthropic key
• update the README environment variable docs so AI_GATEWAY_API_KEY is documented as the key for Claude
• remove stale unused Claude/Anthropic key requirement constants from the task form

Why

The current implementation already executes Claude through AI Gateway, but the template docs and settings UI still imply that Claude needs ANTHROPIC_API_KEY. That creates a broken first-run experience and makes the public template page misleading.

Validation

• pnpm lint
• pnpm type-check
• pnpm build

Impact

Users testing the template should now be prompted for AI Gateway when using Claude, and the repo docs better match the actual runtime behavior.