### 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 = None
        if isinstance(last_automated_dagrun, datetime):
            warnings.warn(
                "Passing a datetime to DAG.next_dagrun_info is deprecated. Use a DataInterval instead.",
                RemovedInAirflow3Warning,
                stacklevel=2,
            )
            data_interval = self.infer_automated_data_interval(
                timezone.coerce_datetime(last_automated_dagrun)
            )
        else:
            data_interval = last_automated_dagrun
        if restricted:
            restriction = self._time_restriction
        else:
            restriction = TimeRestriction(earliest=None, latest=None, catchup=True)
        try:
            info = self.timetable.next_dagrun_info(
                last_automated_data_interval=data_interval,
                restriction=restriction,
            )
        except Exception:
            self.log.exception(
                "Failed to fetch run info after data interval %s for DAG %r",
                data_interval,
                self.dag_id,
            )
            info = None
        return info
    def next_dagrun_after_date(self, date_last_automated_dagrun: pendulum.DateTime | None):