### 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)

"""
    Run a single task instance, and push result to Xcom for downstream tasks.
    Bypasses a lot of extra steps used in `task.run` to keep our local running as fast as
    possible.  This function is only meant for the `dag.test` function as a helper function.
    Args:
        ti: TaskInstance to run
    """
    log.info("*****************************************************")
    if ti.map_index > 0:
        log.info("Running task %s index %d", ti.task_id, ti.map_index)
    else:
        log.info("Running task %s", ti.task_id)
    try:
        ti._run_raw_task(session=session)
        session.flush()
        log.info("%s ran successfully!", ti.task_id)
    except AirflowSkipException:
        log.info("Task Skipped, continuing")
    log.info("*****************************************************")
def _get_or_create_dagrun(
    dag: DAG,
    conf: dict[Any, Any] | None,
    start_date: datetime,
    execution_date: datetime,
    run_id: str,
    session: Session,