docs: update custom models config to use settings.json format

[factory-ben]

Summary

Updates BYOK documentation to reflect the new custom model configuration format introduced in PR #8753.

Changes

• New primary format: Custom models are now configured in ~/.factory/settings.json using camelCase field names
• Field name changes:
  • custom_models → customModels
  • model_display_name → displayName
  • base_url → baseUrl
  • api_key → apiKey
  • max_tokens → maxOutputTokens
  • supports_images → supportsImages
  • extra_args → extraArgs
  • extra_headers → extraHeaders
• Backwards compatibility note: Added documentation about legacy config.json support
• File watching: Settings changes are detected automatically (no restart needed)

Files Updated

• docs/cli/byok/overview.mdx - Main BYOK reference
• docs/cli/configuration/byok.mdx - Configuration BYOK page
• docs/cli/configuration/settings.mdx - Added customModels to settings table
• docs/cli/byok/*.mdx - All provider-specific pages
• docs/cli/droid-exec/overview.mdx - Exec custom model examples
• docs/enterprise/models-llm-gateways-and-integrations.mdx - Enterprise reference

Related

• Implementation PR: https://github.com/Factory-AI/factory-mono/pull/8753

[factory-droid]

Droid finished @factory-ben's task —— View job

---

✅ Code Review Complete

No high-severity issues found in this documentation PR.

Summary of changes reviewed:

• Updates configuration file reference from ~/.factory/config.json to ~/.factory/settings.json
• Migrates field naming convention from snake_case (custom_models, base_url, api_key, max_tokens) to camelCase (customModels, baseUrl, apiKey, maxOutputTokens)
• Adds legacy support note for backwards compatibility with the old format
• Updates the settings.mdx reference table to include customModels
• Fixes the model reference format in droid-exec overview to be consistent (spaces → hyphens)

All JSON examples are syntactically valid and the documentation is consistent across all updated files.