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

.claude/skills/write-e2e-test/SKILL.md

@@ -1,9 +1,9 @@
 ---
 name: write-e2e-test
-description: Write or change a ChatterMate Playwright end-to-end test (e2e/). Use when adding e2e coverage for a feature, fixing a failing e2e spec, or adding a page object, fixture or fake-LLM script.
+description: Write or change a ReplyPier Playwright end-to-end test (e2e/). Use when adding e2e coverage for a feature, fixing a failing e2e spec, or adding a page object, fixture or fake-LLM script.
 ---
 
-# Writing a ChatterMate e2e test
+# Writing a ReplyPier e2e test
 
 The suite lives in `e2e/`. Read `e2e/AUTHORING.md` first — it is the source of truth for
 layout and rules. The essentials: