### Apache Airflow version

Other Airflow 2 version (please specify below)

### What happened

Airflow 2.3.3

Raising the `AirflowSkipException` in the source code, using the `DockerOperator`, is supposed to mark the task as skipped, according to the [docs](https://airflow.apache.org/docs/apache-airflow/stable/concepts/tasks.html#special-exceptions). However, what happens is that the task is marked as failed with the logs showing `ERROR - Task failed with exception`.

### What you think should happen instead

Tasks should be marked as skipped, not failed.

### How to reproduce

Raise the `AirflowSkipException` in the python source code, while using the `DockerOperator`.

### Operating System

Ubuntu 20.04.5 LTS (GNU/Linux 5.4.0-125-generic x86_64)

### Versions of Apache Airflow Providers

_No response_

### Deployment

Docker-Compose

### 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)

) -> None:
        super().__init__(**kwargs)
        self.api_version = api_version
        if type(auto_remove) == bool:
            warnings.warn(
                "bool value for auto_remove is deprecated, please use 'never', 'success', or 'force' instead",
                DeprecationWarning,
                stacklevel=2,
            )
        if str(auto_remove) == "False":
            self.auto_remove = "never"
        elif str(auto_remove) == "True":
            self.auto_remove = "success"
        elif str(auto_remove) in ("never", "success", "force"):
            self.auto_remove = auto_remove
        else:
            raise ValueError("unsupported auto_remove option, use 'never', 'success', or 'force' instead")
        self.command = command
        self.container_name = container_name
        self.cpus = cpus
        self.dns = dns
        self.dns_search = dns_search
        self.docker_url = docker_url
        self.environment = environment or {}
        self._private_environment = private_environment or {}
        self.env_file = env_file
        self.force_pull = force_pull
        self.image = image
        self.mem_limit = mem_limit
        self.host_tmp_dir = host_tmp_dir