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>

tests/test_coordinator.py

@@ -804,3 +804,27 @@ async def test_snapshot_from_api_payload_empty_household() -> None:
     assert snapshot.children == ()
     assert snapshot.cycle_day == 0
     assert snapshot.today_iso == "2026-09-23"
+
+
+#: The panel snapshot exactly as an API from before the ``timezone`` field
+#: serves it: an integration upgraded ahead of its API still reads this.
+LEGACY_SNAPSHOT = {
+    key: value for key, value in FIXTURE_SNAPSHOT.items() if key != "timezone"
+}
+
+
+async def test_coordinator_reads_a_legacy_payload_without_timezone(
+    hass, make_entry
+) -> None:
+    """An older API omits ``timezone``: every refresh still builds the
+    snapshot (the cards then fall back to Home Assistant's zone)."""
+    assert "timezone" not in LEGACY_SNAPSHOT
+    client = StubSnapshotClient(LEGACY_SNAPSHOT)
+    _entry, coordinator = await _wire_entry(hass, make_entry, client)
+    assert coordinator.last_update_success is True
+
+    snapshot = await coordinator._async_update_data()
+    assert snapshot.timezone == ""
+    assert snapshot.cycle_day == 3
+    assert [child.child_name for child in snapshot.children] == ["Ada", "Bo"]
+    assert snapshot_from_api_payload(LEGACY_SNAPSHOT).timezone == ""