### 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)

result.update(
                p_dag._get_task_instances(
                    task_ids=task_ids,
                    start_date=start_date,
                    end_date=end_date,
                    run_id=None,
                    state=state,
                    include_subdags=include_subdags,
                    include_parentdag=False,
                    include_dependent_dags=include_dependent_dags,
                    as_pk_tuple=True,
                    exclude_task_ids=exclude_task_ids,
                    session=session,
                    dag_bag=dag_bag,
                    recursion_depth=recursion_depth,
                    max_recursion_depth=max_recursion_depth,
                    visited_external_tis=visited_external_tis,
                )
            )
        if include_dependent_dags:
            
            from airflow.sensors.external_task import ExternalTaskMarker
            query = tis
            if as_pk_tuple:
                all_tis = session.execute(query).all()
                condition = TI.filter_for_tis(TaskInstanceKey(*cols) for cols in all_tis)
                if condition is not None:
                    query = select(TI).where(condition)
            if visited_external_tis is None:
                visited_external_tis = set()