### 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)

dag = DAG("test_schedule_interval_description", timetable=timetable)
        assert dag.timetable == timetable
        assert dag.timetable.description == expected_description
    def test_create_dagrun_run_id_is_generated(self):
        dag = DAG(dag_id="run_id_is_generated")
        dr = dag.create_dagrun(run_type=DagRunType.MANUAL, execution_date=DEFAULT_DATE, state=State.NONE)
        assert dr.run_id == f"manual__{DEFAULT_DATE.isoformat()}"
    def test_create_dagrun_run_type_is_obtained_from_run_id(self):
        dag = DAG(dag_id="run_type_is_obtained_from_run_id")
        dr = dag.create_dagrun(run_id="scheduled__", state=State.NONE)
        assert dr.run_type == DagRunType.SCHEDULED
        dr = dag.create_dagrun(run_id="custom_is_set_to_manual", state=State.NONE)
        assert dr.run_type == DagRunType.MANUAL
    def test_create_dagrun_job_id_is_set(self):