### 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)

:param dag_hash: Hash of Serialized DAG
        :param data_interval: Data interval of the DagRun
        """
        logical_date = timezone.coerce_datetime(execution_date)
        if data_interval and not isinstance(data_interval, DataInterval):
            data_interval = DataInterval(*map(timezone.coerce_datetime, data_interval))
        if data_interval is None and logical_date is not None:
            warnings.warn(
                "Calling `DAG.create_dagrun()` without an explicit data interval is deprecated",
                RemovedInAirflow3Warning,
                stacklevel=3,
            )
            if run_type == DagRunType.MANUAL:
                data_interval = self.timetable.infer_manual_data_interval(run_after=logical_date)
            else:
                data_interval = self.infer_automated_data_interval(logical_date)
        if run_type is None or isinstance(run_type, DagRunType):
            pass
        elif isinstance(run_type, str):  
            run_type = DagRunType(run_type)
        else:
            raise ValueError(f"`run_type` should be a DagRunType, not {type(run_type)}")
        if run_id:  
            if not isinstance(run_id, str):
                raise ValueError(f"`run_id` should be a str, not {type(run_id)}")
            inferred_run_type = DagRunType.from_run_id(run_id)
            if run_type is None:
                
                run_type = inferred_run_type
            elif run_type == DagRunType.MANUAL and inferred_run_type != DagRunType.MANUAL: