### Summary

The ansible-core 2.15 ignores handlers notified by another handlers when are more than one  case.

With the ansible-core 2.14, all cases of handlers notified by another handlers are runned.

### Issue Type

Bug Report

### Component Name

notify

### Ansible Version

```console
ansible [core 2.15.0]
  config file = /etc/ansible/ansible.cfg
  configured module search path = ['/home/user/.ansible/plugins/modules', '/usr/share/ansible/plugins/modules']
  ansible python module location = /usr/lib/python3.11/site-packages/ansible
  ansible collection location = /home/user/.ansible/collections:/usr/share/ansible/collections
  executable location = /usr/bin/ansible
  python version = 3.11.3 (main, Apr  5 2023, 15:52:25) [GCC 12.2.1 20230201] (/usr/bin/python)
  jinja version = 3.1.2
  libyaml = True
```


### Configuration

```console
CONFIG_FILE() = /etc/ansible/ansible.cfg
EDITOR(env: EDITOR) = vim
PAGER(env: PAGER) = less
```


### OS / Environment

Archlinux

ansible 7.6.0-1
ansible-core 2.15.0-1
ansible-lint 6.15.0.r45.g2fca3fe-2
python-ansible-compat 4.0.2-1

### Steps to Reproduce

<!--- Paste example playbooks or commands between quotes below -->
```yaml (paste below)
---
- name: Testing
  hosts: localhost
  tasks:
    - name: Trigger handlers
      debug:
        msg: Task 1
      changed_when: true
      notify: Handler 1
  handlers:
    - name: Handler 1
      debug:
        msg: Handler 1
      changed_when: true
      notify: Handler 2
    - name: Handler 2
      debug:
        msg: Handler 2
      changed_when: true
      notify: Handler 3
    - name: Handler 3
      debug:
        msg: Handler 3
```


### Expected Results

All handlers must be notified.

```console
PLAY [Testing] ********************************************************************************************************

TASK [Gathering Facts] ********************************************************************************************************
ok: [localhost]

TASK [Trigger handlers] ********************************************************************************************************
changed: [localhost] => {
    "msg": "Task 1"
}

RUNNING HANDLER [Handler 1] ********************************************************************************************************
changed: [localhost] => {
    "msg": "Handler 1"
}

RUNNING HANDLER [Handler 2] ********************************************************************************************************
changed: [localhost] => {
    "msg": "Handler 2"
}

RUNNING HANDLER [Handler 3] ********************************************************************************************************
changed: [localhost] => {
    "msg": "Handler 3"
}

PLAY RECAP ********************************************************************************************************
localhost                  : ok=5    changed=4    unreachable=0    failed=0    skipped=0    rescued=0    ignored=0
```

### Actual Results

Only the two first handlers were notified.

```console
PLAY [Testing] *************************************************************************************************

TASK [Gathering Facts] *************************************************************************************************
ok: [localhost]

TASK [Trigger handlers] *************************************************************************************************
changed: [localhost] => {
    "msg": "Task 1"
}

RUNNING HANDLER [Handler 1] *************************************************************************************************
changed: [localhost] => {
    "msg": "Handler 1"
}

RUNNING HANDLER [Handler 2] *************************************************************************************************
changed: [localhost] => {
    "msg": "Handler 2"
}

PLAY RECAP *************************************************************************************************
localhost                  : ok=4    changed=3    unreachable=0    failed=0    skipped=0    rescued=0    ignored=0
```


### Code of Conduct

- [X] I agree to follow the Ansible Code of Conduct
iterator.add_notification(original_host.name, notification)
                                display.vv(f"Notification for handler {notification} has been saved.")
                                continue
                            msg = (
                                f"The requested handler '{notification}' was not found in either the main handlers"
                                " list nor in the listening handlers list"
                            )
                            if C.ERROR_ON_MISSING_HANDLER:
                                raise AnsibleError(msg)
                            else:
                                display.warning(msg)
                    if 'add_host' in result_item:
                        
                        new_host_info = result_item.get('add_host', dict())
                        self._inventory.add_dynamic_host(new_host_info, result_item)
                        
                        if result_item.get('changed') and new_host_info['host_name'] not in self._hosts_cache_all:
                            self._hosts_cache_all.append(new_host_info['host_name'])
                    elif 'add_group' in result_item:
                        
                        self._inventory.add_dynamic_group(original_host, result_item)
                    if 'add_host' in result_item or 'add_group' in result_item:
                        item_vars = _get_item_vars(result_item, original_task)
                        found_task_vars = self._queued_task_cache.get((original_host.name, task_result._task._uuid))['task_vars']
                        if item_vars:
                            all_task_vars = combine_vars(found_task_vars, item_vars)
                        else:
                            all_task_vars = found_task_vars
                        all_task_vars[original_task.register] = wrap_var(result_item)
                        post_process_whens(result_item, original_task, handler_templar, all_task_vars)