How to observe changes in StateNotifier correctly

state_notifier_test.dart

import 'package:flutter_riverpod/all.dart';
import 'package:flutter_test/flutter_test.dart';

void main() {
  group('StateNotifier', () {
    test('should have correct starting value', () {
      final container = ProviderContainer();

      expect(container.read(testProvider.state), 'init');
    });
    test('should return updated value', () {
      final container = ProviderContainer();

      container.read(testProvider).update();
      expect(container.read(testProvider.state), 'update');
    });
    test('should return updated value when read incorrectly', () {
      final container = ProviderContainer();

      container.read(testProvider).update();
      expect(container.read(testProvider).state, 'update');
    });
    test('shouldn\'t trigger update when read incorrectly', () {
      final container = ProviderContainer();

      expect(container.read(incorrectProvider), 'init');
      container.read(testProvider).update();
      expect(container.read(incorrectProvider), 'init');
    });
    test('should trigger update when read correctly', () {
      final container = ProviderContainer();

      expect(container.read(correctProvider), 'init');
      container.read(testProvider).update();
      expect(container.read(correctProvider), 'update');
    });
  });
}

class TestStateNotifier extends StateNotifier<String> {
  TestStateNotifier() : super('init');

  void update() {
    this.state = 'update';
  }
}

final testProvider = StateNotifierProvider((ref) => TestStateNotifier());
final incorrectProvider = Provider<String>((ref) {
  return ref.watch(testProvider).state;
});
final correctProvider = Provider<String>((ref) {
  return ref.watch(testProvider.state);
});