### 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)

"""
        Triggers on_failure_callback or on_success_callback as appropriate.
        This method gets the context of a single TaskInstance part of this DagRun
        and passes that to the callable along with a 'reason', primarily to
        differentiate DagRun failures.
        .. note: The logs end up in
            ``$AIRFLOW_HOME/logs/scheduler/latest/PROJECT/DAG_FILE.py.log``
        :param dagrun: DagRun object
        :param success: Flag to specify if failure or success callback should be called
        :param reason: Completion reason
        :param session: Database session
        """
        callbacks = self.on_success_callback if success else self.on_failure_callback
        if callbacks:
            callbacks = callbacks if isinstance(callbacks, list) else [callbacks]
            tis = dagrun.get_task_instances(session=session)
            ti = tis[-1]  
            ti.task = self.get_task(ti.task_id)
            context = ti.get_template_context(session=session)
            context.update({"reason": reason})
            for callback in callbacks:
                self.log.info("Executing dag callback function: %s", callback)
                try:
                    callback(context)
                except Exception:
                    self.log.exception("failed to invoke dag state update callback")
                    Stats.incr("dag.callback_exceptions", tags={"dag_id": dagrun.dag_id})
    def get_active_runs(self):