### 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)

warnings.warn(
                "Param `schedule_interval` is deprecated and will be removed in a future release. "
                "Please use `schedule` instead. ",
                RemovedInAirflow3Warning,
                stacklevel=2,
            )
        if timetable is not None:
            warnings.warn(
                "Param `timetable` is deprecated and will be removed in a future release. "
                "Please use `schedule` instead. ",
                RemovedInAirflow3Warning,
                stacklevel=2,
            )
        self.timetable: Timetable
        self.schedule_interval: ScheduleInterval
        self.dataset_triggers: Collection[Dataset] = []
        if isinstance(schedule, Collection) and not isinstance(schedule, str):
            from airflow.datasets import Dataset
            if not all(isinstance(x, Dataset) for x in schedule):
                raise ValueError("All elements in 'schedule' should be datasets")
            self.dataset_triggers = list(schedule)
        elif isinstance(schedule, Timetable):
            timetable = schedule
        elif schedule is not NOTSET:
            schedule_interval = schedule
        if self.dataset_triggers:
            self.timetable = DatasetTriggeredTimetable()
            self.schedule_interval = self.timetable.summary
        elif timetable:
            self.timetable = timetable