feat(train): where sync stands in Settings, with Sync now, and the cog's dot

As Learn's and Todo's: halkyon_app's Sync card first in Settings, and the
dot on the cog while sync fails; the person's SyncService is now part of
TrainPerson, which was the only reason Train had neither (the developer's
review of 3 Nov 2026).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Signed-off-by: Tizian 「ティツィアーン」 <raisondetredev@pm.me>

apps/train/halkyon_train_flutter/test/app/train_app_test.dart

@@ -191,6 +191,10 @@ void main() {
             opened.add(person);
             return TrainPerson(
               person,
+              sync: SyncService(
+                ({required onMerge}) async {},
+                debounce: const Duration(days: 1),
+              ),
               workouts:
                   workouts ??
                   WorkoutRepository(
@@ -488,14 +492,15 @@ void main() {
       }
     });
 
-    testWidgets('Settings groups the training first, then the account, About '
-        'Halkyon Train last and the version small at the very foot', (
-      tester,
-    ) async {
+    testWidgets('Settings groups where sync stands first, with Sync now, then '
+        'the training, then the account, About Halkyon Train last and the '
+        'version small at the very foot', (tester) async {
       await start(tester, usedBefore: true);
       await openSettings(tester);
 
+      expect(find.text('Sync now'), findsOneWidget);
       final order = [
+        'Sync',
         'Your body',
         'Account',
         'Sign out',