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/scripts/upgrade-check.sh

@@ -13,8 +13,8 @@ set -euo pipefail
 E2E_DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")/.." && pwd)"
 REPO_DIR="$(cd "$E2E_DIR/.." && pwd)"
 PG_URL="${E2E_PG_URL:-postgresql://${USER}@127.0.0.1:5432}"; PG_URL="${PG_URL%/}"
-DB_NAME="chattermate_upgrade_e2e"
-FRESH_DB="chattermate_fresh_e2e"
+DB_NAME="replypier_upgrade_e2e"
+FRESH_DB="replypier_fresh_e2e"
 BASE_REF="${1:-$(git -C "$REPO_DIR" describe --tags --abbrev=0 --match 'v*')}"
 if [[ -z "${E2E_PYTHON:-}" ]]; then
   if [[ -x "$REPO_DIR/backend/venv/bin/python" ]]; then E2E_PYTHON="$REPO_DIR/backend/venv/bin/python"; else E2E_PYTHON="python3"; fi