### 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)

"""Create and return an DockerHook (cached)."""
        return self.hook
    @property
    def cli(self) -> APIClient:
        return self.hook.api_client
    def _run_image(self) -> list[str] | str | None:
        """Run a Docker container with the provided image"""
        self.log.info("Starting docker container from image %s", self.image)
        if self.mount_tmp_dir:
            with TemporaryDirectory(prefix="airflowtmp", dir=self.host_tmp_dir) as host_tmp_dir_generated:
                tmp_mount = Mount(self.tmp_dir, host_tmp_dir_generated, "bind")
                try:
                    return self._run_image_with_mounts(self.mounts + [tmp_mount], add_tmp_variable=True)
                except APIError as e:
                    if host_tmp_dir_generated in str(e):
                        self.log.warning(
                            "Using remote engine or docker-in-docker and mounting temporary "
                            "volume from host is not supported. Falling back to "
                            "`mount_tmp_dir=False` mode. You can set `mount_tmp_dir` parameter"
                            " to False to disable mounting and remove the warning"
                        )
                        return self._run_image_with_mounts(self.mounts, add_tmp_variable=False)
                    raise
        else:
            return self._run_image_with_mounts(self.mounts, add_tmp_variable=False)
    def _run_image_with_mounts(self, target_mounts, add_tmp_variable: bool) -> list[str] | str | None: