### 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
connection = plugin_loader.connection_loader.get(play_context.connection, play_context, os.devnull)
                connection.set_options(task_keys=task.dump_attrs(), var_options=all_vars)
                play_context.set_attributes_from_plugin(connection)
            if connection:
                try:
                    connection.reset()
                    msg = 'reset connection'
                except ConnectionError as e:
                    
                    display.debug("got an error while closing persistent connection: %s" % e)
            else:
                msg = 'no connection, nothing to reset'
        else:
            raise AnsibleError("invalid meta action requested: %s" % meta_action, obj=task._ds)
        result = {'msg': msg}
        if skipped:
            result['skipped'] = True
            result['skip_reason'] = skip_reason
        else:
            result['changed'] = False
        if not task.implicit:
            header = skip_reason if skipped else msg
            display.vv(f"META: {header}")
        if isinstance(task, Handler):
            task.remove_host(target_host)
        res = TaskResult(target_host, task, result)
        if skipped:
            self._tqm.send_callback('v2_runner_on_skipped', res)
        return [res]
    def _get_cached_role(self, task, play):