### 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)

"""Exception raised when a model populates data interval fields incorrectly.
    The data interval fields should either both be None (for runs scheduled
    prior to AIP-39), or both be datetime (for runs scheduled after AIP-39 is
    implemented). This is raised if exactly one of the fields is None.
    """
    _template = (
        "Inconsistent {cls}: {start[0]}={start[1]!r}, {end[0]}={end[1]!r}, "
        "they must be either both None or both datetime"
    )
    def __init__(self, instance: Any, start_field_name: str, end_field_name: str) -> None:
        self._class_name = type(instance).__name__
        self._start_field = (start_field_name, getattr(instance, start_field_name))
        self._end_field = (end_field_name, getattr(instance, end_field_name))
    def __str__(self) -> str:
        return self._template.format(cls=self._class_name, start=self._start_field, end=self._end_field)
def _get_model_data_interval(
    instance: Any,
    start_field_name: str,
    end_field_name: str,
) -> DataInterval | None:
    start = timezone.coerce_datetime(getattr(instance, start_field_name))
    end = timezone.coerce_datetime(getattr(instance, end_field_name))
    if start is None:
        if end is not None:
            raise InconsistentDataInterval(instance, start_field_name, end_field_name)
        return None
    elif end is None:
        raise InconsistentDataInterval(instance, start_field_name, end_field_name)
    return DataInterval(start, end)
def create_timetable(interval: ScheduleIntervalArg, timezone: Timezone) -> Timetable: