Publish the API's effective timezone and tolerate an older snapshot payload

TZ-002: with no zone stored the API reads its clock in the host's local
time, but the panel fell back to Home Assistant's zone and the admin app
to the browser's, so dates could disagree near midnight. The API now
names the zone it actually uses (api/host_zone.py: the stored zone, else
the host's own IANA zone from TZ, /etc/localtime or /etc/timezone) and
publishes it as the panel snapshot's `timezone` and the admin settings'
read-only `effective_timezone`. Both cards and the admin app format in
that one zone; Home Assistant's or the browser's zone applies only when
none is published.

TZ-003: snapshot_from_api_payload reads `timezone` with a "" default, so
an integration upgraded ahead of its API keeps refreshing against the
pre-change payload.

Task 7e173881-fb99-426f-a2f0-2678439ffc39

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

api/routes_admin.py

@@ -170,7 +170,10 @@ are deliberately not unified here:
   notify target, the three notification times, the four enable
   toggles, and the ``timezone`` — the IANA household zone the
   household-local dates derive from, empty meaning host local; only an
-  admin's ``PATCH`` ever sets it) — the defaults on a fresh database.
+  admin's ``PATCH`` ever sets it) — the defaults on a fresh database —
+  plus the read-only ``effective_timezone`` (:mod:`api.host_zone`): the
+  zone the API actually reads its clock in, which the admin app formats
+  local time in.
 - ``PATCH /settings`` updates ONLY the supplied fields: the body is a
   JSON object of settings field names to new values, passed STRAIGHT
   to :func:`nestquest_core.settings_store.update_settings` — the
@@ -349,6 +352,7 @@ from pydantic import (
     model_validator,
 )
 
+from api import host_zone
 from api import transitions as api_transitions
 from api.auth import require_admin
 from api.database import DatabaseState
@@ -920,9 +924,14 @@ class AdminSettingsResponse(BaseModel):
     afternoon_reminder_enabled: bool
     end_of_day_report_enabled: bool
     celebration_enabled: bool
-    #: The household IANA time zone the API reads its clock in; ``""``
-    #: means the API host's local time.
+    #: The household IANA time zone the admin stored; ``""`` means the
+    #: API host's local time.
     timezone: str
+    #: The zone the API actually reads its clock in — ``timezone``, or
+    #: the API host's own zone when that is ``""`` (``""`` only when the
+    #: host's zone cannot be named).  Read-only; the admin app formats
+    #: local time in it.
+    effective_timezone: str
 
 
 class AdminMissedSweepResponse(BaseModel):
@@ -2179,8 +2188,9 @@ def _settings_response(
     """Serialize one core NestQuestSettings into the documented payload.
 
     Explicit field-by-field (not ``dataclasses.asdict``) so the payload
-    stays pinned to the eleven documented fields even if the core
-    dataclass later grows another one.
+    stays pinned to the eleven documented fields (plus the read-only
+    ``effective_timezone``) even if the core dataclass later grows
+    another one.
     """
     return AdminSettingsResponse(
         horizon_days=settings.horizon_days,
@@ -2194,6 +2204,7 @@ def _settings_response(
         end_of_day_report_enabled=settings.end_of_day_report_enabled,
         celebration_enabled=settings.celebration_enabled,
         timezone=settings.timezone,
+        effective_timezone=host_zone.effective_timezone(settings.timezone),
     )