### Body

Our CI generates `--package-filter` list in selective checks in order to pass them to `build-docs` and `publish-docs` commands. However after implementing #33876 and #34085 we could switch both commands to use short package ids which we already have (almost - there should be a slight modification for `apache-airflow`, `apache-airflow-providers` and `helm-chart` most likely. This should also simplify some of the unit tests we have.



### Committer

- [X] I acknowledge that I am a maintainer/committer of the Apache Airflow project.
packages.append("apache-airflow")
        if any(file.startswith("docs/apache-airflow-providers/") for file in self._files):
            packages.append("apache-airflow-providers")
        if any(file.startswith(("chart/", "docs/helm-chart")) for file in self._files):
            packages.append("helm-chart")
        if any(file.startswith("docs/docker-stack/") for file in self._files):
            packages.append("docker-stack")
        if providers_affected:
            for provider in providers_affected:
                packages.append(f"apache-airflow-providers-{provider.replace('.', '-')}")
        return " ".join([f"--package-filter {package}" for package in packages])
    @cached_property
    def skip_pre_commits(self) -> str:
        return (
            "identity"
            if self._default_branch == "main"
            else "identity,check-airflow-provider-compatibility,"
            "check-extra-packages-references,check-provider-yaml-valid"
        )
    @cached_property
    def skip_provider_tests(self) -> bool:
        if self._default_branch != "main":
            return True
        if self.full_tests_needed:
            return False
        if any(test_type.startswith("Providers") for test_type in self._get_test_types_to_run()):
            return False
        return True
    @cached_property
    def cache_directive(self) -> str: