refactor: split `RequestContext` between server and client #451

Job	Run time
checks / readme-snippets	10s
checks / pre-commit	39s
checks / test (3.11, locked, ubuntu-latest)	33s
checks / test (3.10, locked, ubuntu-latest)	29s
checks / test (3.12, lowest-direct, windows-latest)	1m 9s
checks / test (3.11, lowest-direct, ubuntu-latest)	31s
checks / test (3.11, lowest-direct, windows-latest)	55s
checks / test (3.10, lowest-direct, ubuntu-latest)	27s
checks / test (3.13, lowest-direct, ubuntu-latest)	51s
checks / test (3.12, locked, windows-latest)	57s
checks / test (3.13, lowest-direct, windows-latest)	2m 47s
checks / test (3.10, lowest-direct, windows-latest)	2m 10s
checks / test (3.12, locked, ubuntu-latest)	33s
checks / test (3.14, locked, ubuntu-latest)	2m 16s
checks / test (3.12, lowest-direct, ubuntu-latest)	35s
checks / test (3.10, locked, windows-latest)	59s
checks / test (3.14, locked, windows-latest)	3m 11s
checks / test (3.14, lowest-direct, windows-latest)	2m 33s
checks / test (3.11, locked, windows-latest)	58s
checks / test (3.13, locked, ubuntu-latest)	47s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 18s
checks / test (3.13, locked, windows-latest)	2m 28s
all-green	3s
	27m 19s

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