docs: Add documentation for stdio server shutdown behavior #1190

Job	Run time
checks / readme-snippets	15s
checks / pre-commit	16s
checks / test (3.11, lowest-direct, windows-latest)	2m 34s
checks / test (3.12, lowest-direct, windows-latest)	3m 38s
checks / test (3.10, locked, windows-latest)	2m 36s
checks / test (3.10, locked, ubuntu-latest)	49s
checks / test (3.11, lowest-direct, ubuntu-latest)	56s
checks / test (3.10, lowest-direct, windows-latest)	2m 15s
checks / test (3.10, lowest-direct, ubuntu-latest)	59s
checks / test (3.11, locked, ubuntu-latest)	49s
checks / test (3.12, locked, ubuntu-latest)	58s
checks / test (3.12, lowest-direct, ubuntu-latest)	58s
checks / test (3.13, lowest-direct, ubuntu-latest)	51s
checks / test (3.13, locked, windows-latest)	2m 38s
checks / test (3.13, lowest-direct, windows-latest)	2m 43s
checks / test (3.13, locked, ubuntu-latest)	47s
checks / test (3.12, locked, windows-latest)	3m 44s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 19s
checks / test (3.11, locked, windows-latest)	2m 29s
checks / test (3.14, lowest-direct, windows-latest)	2m 39s
checks / test (3.14, locked, ubuntu-latest)	2m 24s
checks / test (3.14, locked, windows-latest)	3m 2s
all-green	5s
	39m 44s

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