### Apache Airflow version

2.6.3

### What happened

When running multiple standalone DAG processors with separate DAG directories using the `--subdir` argument the processors keep deactivating each other's DAGs (and reactivating their own).

After stepping through the code with a debugger I think the issue is that the calls [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L794) and [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L798) have no awareness of the DAG directories.

### What you think should happen instead

The DAG processors should not touch each other's DAGs in the metadata DB.

### How to reproduce

Start two or more standalone DAG processors with separate DAG directories and observe (e.g. via the UI) how the list of active DAGs keeps changing constantly.

### Operating System

Linux 94b223524983 6.1.32-0-virt #1-Alpine SMP PREEMPT_DYNAMIC Mon, 05 Jun 2023 09:39:09 +0000 x86_64 x86_64 x86_64 GNU/Linux

### Versions of Apache Airflow Providers

_No response_

### Deployment

Other Docker-based deployment

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

@mock.patch("airflow.dag_processing.processor.DagFileProcessorProcess.pid", new_callable=PropertyMock)
    @mock.patch("airflow.dag_processing.processor.DagFileProcessorProcess")
    def test_kill_timed_out_processors_no_kill(self, mock_dag_file_processor, mock_pid):
        mock_pid.return_value = 1234
        manager = DagProcessorJobRunner(
            job=Job(),
            processor=DagFileProcessorManager(
                dag_directory=TEST_DAG_FOLDER,
                max_runs=1,
                processor_timeout=timedelta(seconds=5),
                signal_conn=MagicMock(),
                dag_ids=[],
                pickle_dags=False,
                async_mode=True,
            ),
        )
        processor = DagFileProcessorProcess(
            file_path="abc.txt",
            pickle_dags=False,
            dag_ids=[],
            dag_directory=str(TEST_DAG_FOLDER),
            callback_requests=[],
        )
        processor._start_time = timezone.make_aware(datetime.max)
        manager.processor._processors = {"abc.txt": processor}
        manager.processor._kill_timed_out_processors()
        mock_dag_file_processor.kill.assert_not_called()
    @conf_vars({("core", "load_examples"): "False"})
    @pytest.mark.execution_timeout(10)
    def test_dag_with_system_exit(self):