docs(samples): add auto-generated samples for Node with api short name in region tag (#221)

- [ ] Regenerate this pull request now.

PiperOrigin-RevId: 399287285

Source-Link: googleapis/googleapis@1575986

Source-Link: https://github.com/googleapis/googleapis-gen/commit/b27fff623a5d8d586b703b5e4919856abe7c2eb3
Copy-Tag: eyJwIjoiLmdpdGh1Yi8uT3dsQm90LnlhbWwiLCJoIjoiYjI3ZmZmNjIzYTVkOGQ1ODZiNzAzYjVlNDkxOTg1NmFiZTdjMmViMyJ9

Author: gcf-owl-bot[bot]

game-servers/snippets/package.json

@@ -4,7 +4,7 @@
   "license": "Apache-2.0",
   "author": "Google LLC",
   "engines": {
-    "node": ">=8"
+    "node": ">=10"
   },
   "files": [
     "*.js"