### 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)

BASE_DATE = timezone.datetime(2022, 7, 20, 20)
        test_dag_id = "test_get_task_instances_before"
        test_task_id = "the_task"
        test_dag = DAG(dag_id=test_dag_id, start_date=BASE_DATE)
        EmptyOperator(task_id=test_task_id, dag=test_dag)
        session = settings.Session()
        def dag_run_before(delta_h=0, type=DagRunType.SCHEDULED):
            dagrun = test_dag.create_dagrun(
                state=State.SUCCESS, run_type=type, run_id=f"test_{delta_h}", session=session
            )
            dagrun.start_date = BASE_DATE + timedelta(hours=delta_h)
            dagrun.execution_date = BASE_DATE + timedelta(hours=delta_h)
            return dagrun
        dr1 = dag_run_before(delta_h=-1, type=DagRunType.MANUAL)  
        dr2 = dag_run_before(delta_h=-2, type=DagRunType.MANUAL)  
        dr3 = dag_run_before(delta_h=-3, type=DagRunType.MANUAL)  
        dr4 = dag_run_before(delta_h=-4, type=DagRunType.MANUAL)  
        dr5 = dag_run_before(delta_h=-5)  
        dr6 = dag_run_before(delta_h=-6)  
        dr7 = dag_run_before(delta_h=-7)  
        dr8 = dag_run_before(delta_h=-8)  
        session.commit()
        REF_DATE = BASE_DATE
        assert set([dr.run_id for dr in [dr1]]) == set(
            [
                ti.run_id
                for ti in test_dag.get_task_instances_before(base_date=REF_DATE, num=1, session=session)
            ]