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

e2e/.env.e2e

@@ -4,7 +4,7 @@
 # NO_ENTERPRISE and the real DATABASE_URL are exported by stack.sh (they depend on
 # the edition and the local Postgres). This placeholder points nowhere, so a backend
 # started without stack.sh fails fast instead of falling back to the dev database.
-DATABASE_URL=postgresql+psycopg://set-by-stack-sh@127.0.0.1:1/chattermate_e2e
+DATABASE_URL=postgresql+psycopg://set-by-stack-sh@127.0.0.1:1/replypier_e2e
 
 ENVIRONMENT=test
 JWT_SECRET_KEY=e2e-jwt-secret-not-for-production
@@ -35,7 +35,7 @@ FIREBASE_CREDENTIALS=/nonexistent/firebase-config.json
 AWS_ACCESS_KEY_ID=
 AWS_SECRET_ACCESS_KEY=
 
-# Providers that ignore the org's base_url (e.g. the CHATTERMATE default model)
+# Providers that ignore the org's base_url (e.g. the REPLYPIER default model)
 # still land on the fake LLM through the openai SDK's env fallback.
 OPENAI_BASE_URL=http://127.0.0.1:14010/v1
 OPENAI_API_KEY=e2e-fake-key