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/tests/core/test_socketio.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.
@@ -81,7 +81,7 @@ def test_configure_socketio_with_redis_enabled(mock_redis_manager, mock_settings
     configure_socketio()
 
     mock_redis_manager.assert_called_once_with(
-        "redis://localhost:6379", write_only=False, channel='chattermate', redis_options=REDIS_OPTIONS,
+        "redis://localhost:6379", write_only=False, channel='replypier', redis_options=REDIS_OPTIONS,
     )
     assert sio.manager is mock_redis_instance
     mock_redis_instance.set_server.assert_called_once_with(sio)
@@ -99,7 +99,7 @@ def test_configure_socketio_with_elasticache(mock_redis_manager, mock_settings,
     configure_socketio()
 
     mock_redis_manager.assert_called_once_with(
-        "rediss://my-cluster.cache.amazonaws.com:6379", write_only=False, channel='chattermate',
+        "rediss://my-cluster.cache.amazonaws.com:6379", write_only=False, channel='replypier',
         redis_options={**REDIS_OPTIONS, 'ssl_cert_reqs': None},
     )