🩹(backend) identify externally provisioned users to PostHog

In the external API, applications authenticate with a
client_id/secret and can provision users with only an email. In that
flow, users are not identified to PostHog, so the user DB id alone
is not enough to identify them in analytics afterwards.

The issue does not exist in the public API, where users are
authenticated and therefore already identified.

Inspired by @flochehab's approach in summary.

CHANGELOG.md

@@ -18,6 +18,10 @@ and this project adheres to
 - 🗑️(settings) deprecate SUMMARY_SERVICE_VERSION=1
 - ⬆️(mail) update mjml to v5 and @html-to/text-cli
 
+### Fixed
+
+- 🩹(backend) identify externally provisioned users to PostHog
+
 ## [1.23.0] - 2026-07-08
 
 ### Added

src/backend/core/external_api/viewsets.py

@@ -215,5 +215,6 @@ class RoomViewSet(
                 "client_id": client_id,
                 "external_api": True,
                 "auth_method": auth_method,
+                "$set": {"email": self.request.user.email},
             },
         )