### 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)

downstream=downstream,
                future=future,
                past=past,
                state=state,
                commit=commit,
                session=session,
            )
            if not commit:
                return altered
            
            
            session.flush()
            task_subset = self.partial_subset(
                task_ids_or_regex=task_ids,
                include_downstream=True,
                include_upstream=False,
            )
            task_subset.clear(
                start_date=start_date,
                end_date=end_date,
                include_subdags=True,
                include_parentdag=True,
                only_failed=True,
                session=session,
                
                exclude_task_ids=frozenset(task_ids),
            )
        return altered
    @property
    def roots(self) -> list[Operator]: