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/FEATURES.md

@@ -21,7 +21,7 @@ One row per user journey. Every feature PR adds or updates its row and the spec
 | Widget chat | Email-first agent (gate → chat, person recorded), Ask anything panel, lead capture with no email asked — each also with the first authorization lost | `specs/widget/chat-styles.spec.ts` | P1 | ✅ |
 | Widget chat | Agent wraps up → customer rates the chat → rating stored against the agent | `specs/widget/rating.spec.ts` | P1 | ✅ |
 | Widget chat | A second browser that types a customer's email gets its own empty conversation (the owner keeps theirs), can't rename them or make them a lead, and an address claimed before its owner signs in yields none of their chats | `specs/widget/conversation-ownership.spec.ts` | P1 | ✅ |
-| Widget chat | Scoping keeps a visitor's own conversation: a second tab joins it; a visitor the site signed in finds it in another browser, and after `ChatterMate.identify()` | `specs/widget/conversation-ownership.spec.ts` | P1 | ✅ |
+| Widget chat | Scoping keeps a visitor's own conversation: a second tab joins it; a visitor the site signed in finds it in another browser, and after `ReplyPier.identify()` | `specs/widget/conversation-ownership.spec.ts` | P1 | ✅ |
 | Widget chat | Provider error, provider timeout and malformed reply each end in an apology; the next message works | `specs/widget/ai-failures.spec.ts` | P1 | ✅ |
 | Human handover | AI hands over → contact form → agent takes over → two-way messages → the AI stays silent | `specs/handover/handover.spec.ts` | P1 | ✅ |
 | Human handover | After hours, or nobody online: the chat waits in the team's queue with a reply that says so | `specs/handover/after-hours.org.spec.ts`, `nobody-online.spec.ts` | P1 | ✅ |