### 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)

are merged into the new *schedule* argument.
    :param dag_id: The id of the DAG; must consist exclusively of alphanumeric
        characters, dashes, dots and underscores (all ASCII)
    :param description: The description for the DAG to e.g. be shown on the webserver
    :param schedule: Defines the rules according to which DAG runs are scheduled. Can
        accept cron string, timedelta object, Timetable, or list of Dataset objects.
        See also :doc:`/howto/timetable`.
    :param start_date: The timestamp from which the scheduler will
        attempt to backfill
    :param end_date: A date beyond which your DAG won't run, leave to None
        for open-ended scheduling
    :param template_searchpath: This list of folders (non-relative)
        defines where jinja will look for your templates. Order matters.
        Note that jinja/airflow includes the path of your DAG file by
        default
    :param template_undefined: Template undefined type.
    :param user_defined_macros: a dictionary of macros that will be exposed
        in your jinja templates. For example, passing ``dict(foo='bar')``
        to this argument allows you to ``{{ foo }}`` in all jinja
        templates related to this DAG. Note that you can pass any
        type of object here.
    :param user_defined_filters: a dictionary of filters that will be exposed
        in your jinja templates. For example, passing
        ``dict(hello=lambda name: 'Hello %s' % name)`` to this argument allows
        you to ``{{ 'world' | hello }}`` in all jinja templates related to
        this DAG.
    :param default_args: A dictionary of default parameters to be used
        as constructor keyword parameters when initialising operators.
        Note that operators have the same hook, and precede those defined
        here, meaning that if your dict contains `'depends_on_past': True`