rename docs/source-app & adjust docs links for lightning

[Borda]

What does this PR do?

seems the deploy was failing bease we have not generated lightning_app
in fact, we shall be changing all imports in unified docs to lightning instead

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cc @Borda @carmocca @akihironitta @justusschock @awaelchli

[github-actions]

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Groups summary

🟢 pytorch_lightning: Docs

Check ID	Status
make-doctest (pytorch)	success	✅
make-html (pytorch)	success	✅

These checks are required after the changes to .github/workflows/docs-checks.yml.

🟡 lightning_app: Tests workflow

Check ID	Status
app-pytest (macOS-11, lightning, 3.8, latest)	no_status	❓
app-pytest (macOS-11, lightning, 3.8, oldest)	no_status	❓
app-pytest (macOS-11, app, 3.9, latest)	no_status	❓
app-pytest (ubuntu-20.04, lightning, 3.8, latest)	no_status	❓
app-pytest (ubuntu-20.04, lightning, 3.8, oldest)	no_status	❓
app-pytest (ubuntu-20.04, app, 3.9, latest)	no_status	❓
app-pytest (windows-2022, lightning, 3.8, latest)	no_status	❓
app-pytest (windows-2022, lightning, 3.8, oldest)	no_status	❓
app-pytest (windows-2022, app, 3.8, latest)	no_status	❓

These checks are required after the changes to tests/tests_app/components/serve/test_model_inference_api.py.

🔴 lightning_app: Docs

Check ID	Status
make-doctest (app)	failure	❌
make-html (app)	failure	❌

These checks are required after the changes to docs/source/Makefile, docs/source/_static/copybutton.js, docs/source/_static/images/icon.svg, docs/source/_static/images/logo-large.svg, docs/source/_static/images/logo-small.svg, docs/source/_static/images/logo.png, docs/source/_static/images/logo.svg, docs/source/_static/main.css, docs/source/_templates/classtemplate.rst, docs/source/_templates/classtemplate_no_index.rst, docs/source/_templates/layout.html, docs/source/_templates/theme_variables.jinja, docs/source/api_reference/components.rst, docs/source/api_reference/core.rst, docs/source/api_reference/frontend.rst, docs/source/api_reference/runners.rst, docs/source/api_reference/storage.rst, docs/source/basics.rst, docs/source/cloud/customize_a_lightning_cluster.rst, docs/source/code_samples/basics/0.py, docs/source/code_samples/basics/1.py, docs/source/code_samples/convert_pl_to_app/app.py, docs/source/code_samples/convert_pl_to_app/requirements.txt, 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[Borda]

@awaelchli, all good catches, and sorry I missed then... :(

UPDATE: redone the whole PR from ground paying proper attention to it 😇

[Borda]

the true check is in: https://github.com/Lightning-AI/lightning/actions/runs/4149219807
@awaelchli could you pls check again 😇

[awaelchli]

can't find this, where was it moved?

[Borda]

This was removed as it duplicated the generated on and caused an issue while building docs, in fact, that shall be removed a while back as this was a bad artifact...

[carmocca]

Is this intended? Do we not want to just include the app files?

[Borda]

unfortunalyly it is inted and it was explicitly asked by @williamFalcon as he was not happy about formating docs...
cc: @justusschock