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

backend/app/api/help_center_public.py

@@ -1,5 +1,5 @@
 """
-Copyright 2024-2026 ChatterMate
+Copyright 2024-2026 ReplyPier
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -74,7 +74,7 @@ from app.services.help_center_seo import (
 )
 from app.services.public_rate_limit import allow_request
 
-public_app = FastAPI(title="ChatterMate Help Center", docs_url=None, redoc_url=None, openapi_url=None)
+public_app = FastAPI(title="ReplyPier Help Center", docs_url=None, redoc_url=None, openapi_url=None)
 install_error_handlers(public_app)
 # Serve ONLY help-center images on this app, so relative logo/article-image paths
 # resolve on a subdomain/custom-domain origin (host dispatch). Scoped to the
@@ -154,11 +154,11 @@ async def _chrome_context(row: HelpCenterSettings, base_path: str = "") -> dict:
         "favicon_url": await resolve_public_url(row.favicon_url) if row.favicon_url else None,
         "header_links": row.header_links or [],
         "widget_id": widget_id_for(row),
-        # The widget LOADER (chattermate.min.js) is served by the frontend, while
+        # The widget LOADER (replypier.min.js) is served by the frontend, while
         # the widget APP it pulls in (/assets/widget.js) is served from
         # VITE_WIDGET_URL by the backend. In prod both resolve to the app domain.
-        "widget_script_url": f"{settings.FRONTEND_URL}/webclient/chattermate.min.js",
-        # This install's API root, handed to the loader as window.chattermateBaseUrl
+        "widget_script_url": f"{settings.FRONTEND_URL}/webclient/replypier.min.js",
+        # This install's API root, handed to the loader as window.replypierBaseUrl
         # (same contract as the dashboard's embed snippet). Without it the loader
         # uses its build-time default — the vendor cloud — which no self-hosted
         # deployment can talk to.