rename integrations_app for accommodating flagships

[Borda]

What does this PR do?

preparation for #16304

Before submitting

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• Did you make sure to update the documentation with your changes? (if necessary)
• Did you write any new necessary tests? (not for typos and docs)
• Did you verify new and existing tests pass locally with your changes?
• Did you list all the breaking changes introduced by this pull request?
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PR review

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Did you have fun?

Make sure you had fun coding 🙃

cc @justusschock @awaelchli @Borda

[github-actions]

⛈️ Required checks status: Has failure 🔴

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Groups summary

🟢 lightning_app: Examples

Check ID	Status
app-examples (macOS-11, app, 3.9, latest)	success	✅
app-examples (macOS-11, app, 3.9, oldest)	success	✅
app-examples (macOS-11, lightning, 3.9, latest)	success	✅
app-examples (ubuntu-20.04, app, 3.9, latest)	success	✅
app-examples (ubuntu-20.04, app, 3.9, oldest)	success	✅
app-examples (ubuntu-20.04, lightning, 3.9, latest)	success	✅
app-examples (windows-2022, app, 3.9, latest)	success	✅
app-examples (windows-2022, app, 3.9, oldest)	success	✅
app-examples (windows-2022, lightning, 3.9, latest)	success	✅

These checks are required after the changes to .github/workflows/ci-examples-app.yml, tests/integrations_app/__init__.py, tests/integrations_app/apps/collect_failures/__init__.py, tests/integrations_app/apps/collect_failures/app.py, tests/integrations_app/apps/collect_failures/requirements.txt, tests/integrations_app/apps/core_features_app/__init__.py, tests/integrations_app/apps/core_features_app/app.py, tests/integrations_app/apps/custom_work_dependencies/__init__.py, tests/integrations_app/apps/custom_work_dependencies/app.py, tests/integrations_app/apps/idle_timeout/__init__.py, tests/integrations_app/apps/idle_timeout/app.py, tests/integrations_app/conftest.py, tests/integrations_app/local/__init__.py, tests/integrations_app/local/test_collect_failures.py, tests/integrations_app/local/test_core_features_app.py, tests/integrations_app/local/test_custom_work_dependencies.py, tests/integrations_app/local/test_idle_timeout.py, tests/integrations_app/public/__init__.py, tests/integrations_app/public/test_app_dag.py, tests/integrations_app/public/test_argparse.py, tests/integrations_app/public/test_boring_app.py, tests/integrations_app/public/test_commands_and_api.py, tests/integrations_app/public/test_drive.py, tests/integrations_app/public/test_gradio.py, tests/integrations_app/public/test_installation_commands_app.py, tests/integrations_app/public/test_layout.py, tests/integrations_app/public/test_multi_node.py, tests/integrations_app/public/test_payload.py, tests/integrations_app/public/test_pickle_or_not.py, tests/integrations_app/public/test_quick_start.py, tests/integrations_app/public/test_scripts.py, tests/integrations_app/public/test_template_react_ui.py, tests/integrations_app/public/test_template_streamlit_ui.py, tests/integrations_app/public/test_v0_app.py.

🔴 lightning_app: Azure

Check ID	Status
App.cloud-e2e	failure	❌

These checks are required after the changes to .azure/app-cloud-e2e.yml, tests/integrations_app/__init__.py, tests/integrations_app/apps/collect_failures/__init__.py, tests/integrations_app/apps/collect_failures/app.py, tests/integrations_app/apps/collect_failures/requirements.txt, tests/integrations_app/apps/core_features_app/__init__.py, tests/integrations_app/apps/core_features_app/app.py, tests/integrations_app/apps/custom_work_dependencies/__init__.py, tests/integrations_app/apps/custom_work_dependencies/app.py, tests/integrations_app/apps/idle_timeout/__init__.py, tests/integrations_app/apps/idle_timeout/app.py, tests/integrations_app/conftest.py, tests/integrations_app/local/__init__.py, tests/integrations_app/local/test_collect_failures.py, tests/integrations_app/local/test_core_features_app.py, tests/integrations_app/local/test_custom_work_dependencies.py, tests/integrations_app/local/test_idle_timeout.py, tests/integrations_app/public/__init__.py, tests/integrations_app/public/test_app_dag.py, tests/integrations_app/public/test_argparse.py, tests/integrations_app/public/test_boring_app.py, tests/integrations_app/public/test_commands_and_api.py, tests/integrations_app/public/test_drive.py, tests/integrations_app/public/test_gradio.py, tests/integrations_app/public/test_installation_commands_app.py, tests/integrations_app/public/test_layout.py, tests/integrations_app/public/test_multi_node.py, tests/integrations_app/public/test_payload.py, tests/integrations_app/public/test_pickle_or_not.py, tests/integrations_app/public/test_quick_start.py, tests/integrations_app/public/test_scripts.py, tests/integrations_app/public/test_template_react_ui.py, tests/integrations_app/public/test_template_streamlit_ui.py, tests/integrations_app/public/test_v0_app.py.

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