refactor: rename web/ to api/ to better reflect its purpose

The web/ folder contains the REST API, WebSocket server, and OAuth
routes — not a web frontend. Renaming to api/ clarifies this distinction
since the actual web frontend lives in panel/.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

api/.gitignore

@@ -0,0 +1,34 @@
+# dependencies (bun install)
+node_modules
+
+# output
+out
+dist
+*.tgz
+
+# code coverage
+coverage
+*.lcov
+
+# logs
+logs
+_.log
+report.[0-9]_.[0-9]_.[0-9]_.[0-9]_.json
+
+# dotenv environment variable files
+.env
+.env.development.local
+.env.test.local
+.env.production.local
+.env.local
+
+# caches
+.eslintcache
+.cache
+*.tsbuildinfo
+
+# IntelliJ based IDEs
+.idea
+
+# Finder (MacOS) folder config
+.DS_Store