refactor: split `RequestContext` between server and client #450

Job	Run time
checks / pre-commit	36s
checks / readme-snippets	11s
checks / test (3.10, lowest-direct, ubuntu-latest)	29s
checks / test (3.11, lowest-direct, ubuntu-latest)	31s
checks / test (3.13, locked, ubuntu-latest)	48s
checks / test (3.11, locked, ubuntu-latest)	28s
checks / test (3.10, locked, ubuntu-latest)	30s
checks / test (3.12, lowest-direct, ubuntu-latest)	32s
checks / test (3.12, lowest-direct, windows-latest)	1m 1s
checks / test (3.12, locked, ubuntu-latest)	32s
checks / test (3.11, locked, windows-latest)	1m 3s
checks / test (3.11, lowest-direct, windows-latest)	1m 0s
checks / test (3.14, lowest-direct, windows-latest)	2m 27s
checks / test (3.13, lowest-direct, windows-latest)	2m 43s
checks / test (3.10, locked, windows-latest)	1m 1s
checks / test (3.14, locked, windows-latest)	3m 29s
checks / test (3.12, locked, windows-latest)	55s
checks / test (3.13, locked, windows-latest)	2m 56s
checks / test (3.13, lowest-direct, ubuntu-latest)	52s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 14s
checks / test (3.10, lowest-direct, windows-latest)	55s
checks / test (3.14, locked, ubuntu-latest)	2m 20s
all-green	5s
	26m 38s

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