### 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)

data_interval: tuple[datetime, datetime] | None = None,
) -> DagRun:
    """Create a DAG run, replacing an existing instance if needed to prevent collisions.
    This function is only meant to be used by :meth:`DAG.test` as a helper function.
    :param dag: DAG to be used to find run.
    :param conf: Configuration to pass to newly created run.
    :param start_date: Start date of new run.
    :param execution_date: Logical date for finding an existing run.
    :param run_id: Run ID for the new DAG run.
    :return: The newly created DAG run.
    """
    log.info("dagrun id: %s", dag.dag_id)
    dr: DagRun = session.scalar(
        select(DagRun).where(DagRun.dag_id == dag.dag_id, DagRun.execution_date == execution_date)
    )
    if dr:
        session.delete(dr)
        session.commit()
    dr = dag.create_dagrun(
        state=DagRunState.RUNNING,
        execution_date=execution_date,
        run_id=run_id,
        start_date=start_date or execution_date,
        session=session,
        conf=conf,
        data_interval=data_interval,
    )
    log.info("created dagrun %s", dr)
    return dr