refactor: split `RequestContext` between server and client #453

Job	Run time
checks / readme-snippets	11s
checks / pre-commit	30s
checks / test (3.10, lowest-direct, windows-latest)	2m 37s
checks / test (3.11, locked, ubuntu-latest)	51s
checks / test (3.11, lowest-direct, ubuntu-latest)	56s
checks / test (3.11, locked, windows-latest)	3m 4s
checks / test (3.12, locked, ubuntu-latest)	53s
checks / test (3.10, locked, ubuntu-latest)	50s
checks / test (3.13, locked, ubuntu-latest)	52s
checks / test (3.12, locked, windows-latest)	4m 1s
checks / test (3.11, lowest-direct, windows-latest)	3m 20s
checks / test (3.10, locked, windows-latest)	2m 42s
checks / test (3.12, lowest-direct, ubuntu-latest)	55s
checks / test (3.13, lowest-direct, windows-latest)	2m 47s
checks / test (3.13, locked, windows-latest)	2m 13s
checks / test (3.12, lowest-direct, windows-latest)	3m 42s
checks / test (3.13, lowest-direct, ubuntu-latest)	55s
checks / test (3.14, lowest-direct, windows-latest)	3m 5s
checks / test (3.14, locked, windows-latest)	3m 28s
checks / test (3.10, lowest-direct, ubuntu-latest)	52s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 21s
checks / test (3.14, locked, ubuntu-latest)	2m 18s
all-green	3s
	42m 26s

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