Update Lightning AI multi-node guide (Trainer)

[awaelchli]

What does this PR do?

Follow up to #19324
The description is the same as in #19324, with only the code snippets and tutorial video changed.

Closes #12143

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📚 Documentation preview 📚: https://pytorch-lightning--19530.org.readthedocs.build/en/19530/

cc @Borda

[github-actions]

⚡ Required checks status: All passing 🟢

Groups summary

🟢 pytorch_lightning: Docs

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

These checks are required after the changes to docs/source-pytorch/clouds/cluster.rst, docs/source-pytorch/clouds/lightning_ai.rst, docs/source-pytorch/common/index.rst, docs/source-pytorch/common_usecases.rst, docs/source-pytorch/levels/intermediate.rst, docs/source-pytorch/levels/intermediate_level_14.rst.

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Thank you for your contribution! 💜

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🔎 Detected hardcoded secrets in your pull request

GitGuardian id	GitGuardian status	Secret	Commit	Filename
5685611	Triggered	Generic High Entropy Secret	403a912	tests/tests_app/utilities/test_login.py	View secret
7296313	Triggered	Base64 Basic Authentication	403a912	tests/tests_app/utilities/test_login.py	View secret

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[lantiga]

Looks great