refactor: split `RequestContext` between server and client #454

Job	Run time
checks / readme-snippets	12s
checks / pre-commit	33s
checks / test (3.10, locked, windows-latest)	2m 40s
checks / test (3.10, locked, ubuntu-latest)	54s
checks / test (3.11, locked, ubuntu-latest)	52s
checks / test (3.12, lowest-direct, windows-latest)	3m 56s
checks / test (3.11, lowest-direct, windows-latest)	2m 42s
checks / test (3.10, lowest-direct, windows-latest)	2m 33s
checks / test (3.10, lowest-direct, ubuntu-latest)	58s
checks / test (3.11, lowest-direct, ubuntu-latest)	55s
checks / test (3.14, locked, ubuntu-latest)	2m 25s
checks / test (3.12, locked, ubuntu-latest)	54s
checks / test (3.11, locked, windows-latest)	2m 31s
checks / test (3.13, lowest-direct, ubuntu-latest)	54s
checks / test (3.14, locked, windows-latest)	5m 17s
checks / test (3.13, lowest-direct, windows-latest)	2m 36s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 17s
checks / test (3.12, lowest-direct, ubuntu-latest)	58s
checks / test (3.13, locked, windows-latest)	2m 33s
checks / test (3.12, locked, windows-latest)	3m 44s
checks / test (3.13, locked, ubuntu-latest)	47s
checks / test (3.14, lowest-direct, windows-latest)	2m 20s
all-green	3s
	42m 34s

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