refactor: split `RequestContext` between server and client (#1987) #455

Job	Run time
checks / readme-snippets	12s
checks / pre-commit	34s
checks / test (3.10, locked, windows-latest)	2m 19s
checks / test (3.10, lowest-direct, windows-latest)	3m 4s
checks / test (3.11, lowest-direct, ubuntu-latest)	56s
checks / test (3.11, lowest-direct, windows-latest)	2m 41s
checks / test (3.12, lowest-direct, windows-latest)	3m 37s
checks / test (3.12, locked, windows-latest)	3m 21s
checks / test (3.10, lowest-direct, ubuntu-latest)	58s
checks / test (3.12, locked, ubuntu-latest)	54s
checks / test (3.14, lowest-direct, windows-latest)	2m 46s
checks / test (3.13, lowest-direct, windows-latest)	2m 35s
checks / test (3.12, lowest-direct, ubuntu-latest)	56s
checks / test (3.13, locked, ubuntu-latest)	44s
checks / test (3.11, locked, ubuntu-latest)	53s
checks / test (3.13, locked, windows-latest)	2m 20s
checks / test (3.14, lowest-direct, ubuntu-latest)	1m 13s
checks / test (3.11, locked, windows-latest)	2m 38s
checks / test (3.14, locked, windows-latest)	3m 25s
checks / test (3.13, lowest-direct, ubuntu-latest)	51s
checks / test (3.10, locked, ubuntu-latest)	47s
checks / test (3.14, locked, ubuntu-latest)	2m 24s
all-green	2s
	40m 10s

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