### 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)

def generate_run_id(self, *, run_type, logical_date, data_interval, **extra) -> str:
            return "abc"
    dag = DAG(dag_id="test", start_date=DEFAULT_DATE, schedule=CustomRunIdTimetable())
    dag_run = dag.create_dagrun(
        run_type=DagRunType.MANUAL,
        state=DagRunState.QUEUED,
        execution_date=DEFAULT_DATE,
        data_interval=(DEFAULT_DATE, DEFAULT_DATE),
    )
    assert dag_run.run_id == "abc"
@pytest.mark.parametrize(
    "run_id_type",
    [DagRunType.BACKFILL_JOB, DagRunType.SCHEDULED, DagRunType.DATASET_TRIGGERED],
)
def test_create_dagrun_disallow_manual_to_use_automated_run_id(run_id_type: DagRunType) -> None:
    dag = DAG(dag_id="test", start_date=DEFAULT_DATE, schedule="@daily")
    run_id = run_id_type.generate_run_id(DEFAULT_DATE)
    with pytest.raises(ValueError) as ctx:
        dag.create_dagrun(
            run_type=DagRunType.MANUAL,
            run_id=run_id,
            execution_date=DEFAULT_DATE,
            data_interval=(DEFAULT_DATE, DEFAULT_DATE),
            state=DagRunState.QUEUED,
        )
    assert str(ctx.value) == (
        f"A manual DAG run cannot use ID {run_id!r} since it is reserved for {run_id_type.value} runs"
    )
class TestTaskClearingSetupTeardownBehavior: