chore: settle product naming across app, widget and deployments

One word everywhere: ReplyPier in UI copy, package and image names, the
replypier.site domains (app/api/docs/help), REPLYPIER_* settings and the
--rp-* design tokens.

- public widget API is window.ReplyPier with replypierId / replypierBaseUrl /
  replypierConfig / replypierToken, loader at webclient/replypier.min.js;
  both bundles rebuilt from source
- NOTICE, llms.txt, CONTRIBUTING, HELP_CENTER_INFRA and the demo runtime
  config rewritten; the demo config ships no third-party keys
- migration adds the ReplyPier managed model value to the AI model enum
- local compose project, database, role, volume and Docker network carry the
  same name

deploy/local/init_env.py

@@ -26,7 +26,7 @@ def main() -> None:
     password = compose_env.read_text().strip().split("=", 1)[1]
     fernet_key = base64.b64encode(base64.urlsafe_b64encode(secrets.token_bytes(32))).decode()
     settings = {
-        "DATABASE_URL": f"postgresql+psycopg://chattermate:{password}@127.0.0.1:15432/chattermate",
+        "DATABASE_URL": f"postgresql+psycopg://replypier:{password}@127.0.0.1:15432/replypier",
         "REDIS_ENABLED": "true",
         "REDIS_URL": "redis://127.0.0.1:16380/0",
         "ENVIRONMENT": "production",