fix: complete the docs/→design/ and openapi→api/ rename references

The file moves (docs/ → design/, docs/api/openapi.json → api/openapi.json) landed
in d01a8fd, but the matching reference updates did not — so mgmt.rs's drift-test
`include_str!("../../../docs/api/openapi.json")` pointed at a path that no longer
exists and the host failed to build. This restores it and updates every reference:

  - mgmt.rs include_str! → ../../../api/openapi.json (fixes the build)
  - web/orval.config.ts codegen target, web/Dockerfile, .dockerignore
  - deb/rpm/Arch packaging install paths
  - CLAUDE.md, the .gitea CI workflows, code doc-comments, design-doc cross-links

docs-site route URLs (/docs/...) untouched.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

clients/apple/README.md

@@ -361,4 +361,4 @@ ever switched to a logged-in GUI session, re-adding macOS to the job's capture s
 - Mid-stream renegotiation (resolution change without reconnect) is designed-for but not
   implemented (the Welcome is one-shot today).
 - Host-side gamepad injection needs `/dev/uinput` access on the box (udev rule from
-  `docs/linux-setup.md`).
+  `design/linux-setup.md`).