### 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)

session.merge(dagrun_1)
        task_instance_1 = TI(t_1, execution_date=DEFAULT_DATE, state=ti_state_begin)
        task_instance_1.job_id = 123
        session.merge(task_instance_1)
        session.commit()
        dag.clear(
            start_date=DEFAULT_DATE,
            end_date=DEFAULT_DATE + datetime.timedelta(days=1),
            session=session,
        )
        task_instances = (
            session.query(
                TI,
            )
            .filter(
                TI.dag_id == dag_id,
            )
            .all()
        )
        assert len(task_instances) == 1
        task_instance: TI = task_instances[0]
        assert task_instance.state == ti_state_end
        self._clean_up(dag_id)
    def test_next_dagrun_info_once(self):
        dag = DAG("test_scheduler_dagrun_once", start_date=timezone.datetime(2015, 1, 1), schedule="@once")
        next_info = dag.next_dagrun_info(None)
        assert next_info and next_info.logical_date == timezone.datetime(2015, 1, 1)
        next_info = dag.next_dagrun_info(next_info.data_interval)
        assert next_info is None
    def test_next_dagrun_info_start_end_dates(self):