docs(examples): Use refactored shinylive syntax #3101

Job	Run time
playwright-examples (3.11, ubuntu-latest)	8m 9s
playwright-examples (3.10, ubuntu-latest)	8m 40s
playwright-examples (3.9, ubuntu-latest)	8m 32s
playwright-examples (3.8, ubuntu-latest)	8m 18s
build (3.11, ubuntu-latest)	2m 7s
build (3.11, windows-latest)	2m 16s
build (3.11, macOS-latest)	2m 4s
playwright-shiny (3.11, ubuntu-latest)	8m 50s
build (3.10, ubuntu-latest)	2m 5s
playwright-shiny (3.10, ubuntu-latest)	8m 53s
build (3.10, windows-latest)	2m 1s
playwright-shiny (3.9, ubuntu-latest)	9m 2s
build (3.10, macOS-latest)	2m 5s
playwright-shiny (3.8, ubuntu-latest)	8m 52s
build (3.9, ubuntu-latest)	2m 19s
build (3.9, windows-latest)	2m 2s
build (3.9, macOS-latest)	2m 8s
build (3.8, ubuntu-latest)	2m 27s
playwright-deploys	0s
build (3.8, windows-latest)	1m 49s
build (3.8, macOS-latest)	1m 48s
testrail-reporting-nightly	0s
Deploy to PyPI	0s
	1h 34m 27s

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