docs: update api-keys-setup.md for Phase 2 (Anthropic now wired up)

Marks Anthropic as live (agent.type: "anthropic" works once a key is
configured), adds the budget-bucket-separation and [runners] disable notes
that Phase 2's implementation agent independently drafted in a since-discarded
duplicate doc, adds an explicit warning against using claude setup-token /
CLI OAuth session for the native adapter (subscription auth is for Claude
Code/Agent SDK use, not a separate product's API calls, per Anthropic's Agent
SDK terms), and bumps default_model examples from claude-sonnet-4-5 to
claude-sonnet-5.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01V1moSNCJRcP6kykA4tyUSs

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