### 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)

"""
        Tests that an attempt to schedule a task after the Dag's end_date
        does not succeed.
        """
        delta = datetime.timedelta(hours=1)
        runs = 3
        start_date = DEFAULT_DATE
        end_date = start_date + (runs - 1) * delta
        dag_id = "test_schedule_dag_start_end_dates"
        dag = DAG(dag_id=dag_id, start_date=start_date, end_date=end_date, schedule=delta)
        dag.add_task(BaseOperator(task_id="faketastic", owner="Also fake"))
        
        dates = []
        interval = None
        for _ in range(runs):
            next_info = dag.next_dagrun_info(interval)
            if next_info is None:
                dates.append(None)
            else:
                interval = next_info.data_interval
                dates.append(interval.start)
        assert all(date is not None for date in dates)
        assert dates[-1] == end_date
        assert dag.next_dagrun_info(interval.start) is None
    def test_next_dagrun_info_catchup(self):