Split `process_client` flag into two flags #14239

Job	Run time
build	2m 12s
build-wasm (web)	4m 13s
deploy-target	12s
build-wasm (nodejs)	3m 34s
stable - x86_64-unknown-linux-gnu - node@16	5m 7s
Deploy examples	0s
Deploy preview tarball	1m 49s
Potentially publish release	0s
Upload Turbopack Bytesize metrics to Datadog	0s
Release Stats	0s
	17m 7s

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