### 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)

"""
        Test to check that a DAG with catchup = False only schedules beginning now, not back to the start date
        """
        def make_dag(dag_id, schedule, start_date, catchup):
            default_args = {
                "owner": "airflow",
                "depends_on_past": False,
            }
            dag = DAG(
                dag_id,
                schedule=schedule,
                start_date=start_date,
                catchup=catchup,
                default_args=default_args,
            )
            op1 = EmptyOperator(task_id="t1", dag=dag)
            op2 = EmptyOperator(task_id="t2", dag=dag)
            op3 = EmptyOperator(task_id="t3", dag=dag)
            op1 >> op2 >> op3
            return dag
        now = timezone.utcnow()
        six_hours_ago_to_the_hour = (now - datetime.timedelta(hours=6)).replace(
            minute=0, second=0, microsecond=0
        )
        half_an_hour_ago = now - datetime.timedelta(minutes=30)
        two_hours_ago = now - datetime.timedelta(hours=2)
        dag1 = make_dag(
            dag_id="dag_without_catchup_ten_minute",