### 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)

self,
        dag: DAG,
        most_recent_dag_run: None | datetime | DataInterval,
    ) -> None:
        """
        Calculate ``next_dagrun`` and `next_dagrun_create_after``.
        :param dag: The DAG object
        :param most_recent_dag_run: DataInterval (or datetime) of most recent run of this dag, or none
            if not yet scheduled.
        """
        most_recent_data_interval: DataInterval | None
        if isinstance(most_recent_dag_run, datetime):
            warnings.warn(
                "Passing a datetime to `DagModel.calculate_dagrun_date_fields` is deprecated. "
                "Provide a data interval instead.",
                RemovedInAirflow3Warning,
                stacklevel=2,
            )
            most_recent_data_interval = dag.infer_automated_data_interval(most_recent_dag_run)
        else:
            most_recent_data_interval = most_recent_dag_run