Rename context.go to client_context.go (#1082)

Here, rename `context.go` to `client_context.go`. The premise here is to
give it a less generic name so it's easier to find via quick file find
when working from memory (something I currently have trouble doing). In
case there's anything that lands in here not related to client-specific
context in the future, we can find a different name for the file, but
it's been only client-specific since inception so far.

Author: brandur