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

Dockerfile.postgres

@@ -18,7 +18,7 @@ RUN git clone https://github.com/pgvector/pgvector.git && \
 COPY ./backend/scripts/init-vector.sql /docker-entrypoint-initdb.d/
 
 # Set environment variables
-ENV POSTGRES_DB=chattermate
+ENV POSTGRES_DB=replypier
 ENV POSTGRES_USER=postgres
 ENV POSTGRES_PASSWORD=postgres