refactor: split `RequestContext` between server and client #452

Job	Run time
checks / pre-commit	39s
checks / readme-snippets	11s
checks / test (3.10, lowest-direct, windows-latest)	2m 28s
checks / test (3.11, lowest-direct, ubuntu-latest)	51s
checks / test (3.10, lowest-direct, ubuntu-latest)	56s
checks / test (3.12, lowest-direct, ubuntu-latest)	55s
checks / test (3.10, locked, windows-latest)	2m 25s
checks / test (3.10, locked, ubuntu-latest)	55s
checks / test (3.12, lowest-direct, windows-latest)	3m 40s
checks / test (3.13, locked, windows-latest)	2m 47s
checks / test (3.13, lowest-direct, windows-latest)	2m 44s
checks / test (3.14, locked, windows-latest)	3m 13s
checks / test (3.14, lowest-direct, windows-latest)	2m 21s
checks / test (3.11, locked, ubuntu-latest)	50s
checks / test (3.12, locked, ubuntu-latest)	57s
checks / test (3.11, locked, windows-latest)	2m 27s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 22s
checks / test (3.11, lowest-direct, windows-latest)	3m 5s
checks / test (3.14, locked, ubuntu-latest)	2m 24s
checks / test (3.12, locked, windows-latest)	3m 31s
checks / test (3.13, locked, ubuntu-latest)	50s
checks / test (3.13, lowest-direct, ubuntu-latest)	53s
all-green	5s
	40m 29s

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