docs: update logs example to use opentelemetry-instrumentation-logging (#5344)

* docs: update logs example to use opentelemetry-instrumentation-logging

* docs: restore experimental warning, keep scope focused on LoggingHandler

* add changelog fragment for #5344

* docs: address review feedback, use deprecated and update propagate comment

* Apply suggestion from @xrmx

* Apply suggestion from @xrmx

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Co-authored-by: Riccardo Magliocchetti <riccardo.magliocchetti@gmail.com>