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/alembic/versions/replypier_ai_model_001_add_managed_model_value.py

@@ -0,0 +1,52 @@
+"""
+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.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+"""
+
+"""Add the ReplyPier managed model to the AI model enum
+
+Revision ID: replypier_ai_model_001
+Revises: widget_key_lookup_001
+Create Date: 2026-10-26 00:00:00.000000
+
+``AIModelType`` carries a managed-provider member for installs that bill model
+usage through ReplyPier instead of the customer's own key. The enum type on an
+existing database does not have that value yet, so add it; a database created
+from the current head already contains it and this is then a no-op.
+"""
+from typing import Sequence, Union
+
+from alembic import op
+
+# revision identifiers, used by Alembic.
+revision: str = 'replypier_ai_model_001'
+down_revision: Union[str, None] = 'widget_key_lookup_001'
+branch_labels: Union[str, Sequence[str], None] = None
+depends_on: Union[str, Sequence[str], None] = None
+
+MANAGED_VALUE = 'REPLYPIER'
+
+
+def upgrade() -> None:
+    # ADD VALUE cannot be used in the same transaction that references the new
+    # label, so it runs in its own autocommit block.
+    with op.get_context().autocommit_block():
+        op.execute(
+            f"ALTER TYPE aimodeltype ADD VALUE IF NOT EXISTS '{MANAGED_VALUE}'"
+        )
+
+
+def downgrade() -> None:
+    # PostgreSQL cannot drop an enum value; nothing to undo.
+    pass