Don’t use request context for sending responses

[andrewbranch]

In #2499, we started checking for context cancelation during sending responses as a way to ensure we don’t get stuck trying to write a message to a client that has exited. If the context is canceled, we would bail out returning an error, but that error would eventually be ignored. Ignoring the error is a problem, but the big problem was that we were checking the request context during sending. If the request was canceled, we would try to send an error response saying the request was canceled, but we would fail to send it since the context was canceled. This left the client perpetually waiting for our response and never asking for new errors.

Fixes #2557

[Copilot]

Pull request overview

This pull request fixes a critical issue where the language server would fail to send error responses when requests were canceled, leaving clients waiting indefinitely. The fix separates the context used for request processing from the context used for sending responses.

Changes:

• Removed context parameters from all send-related functions (send, sendResult, sendError, sendResponse), using the server's backgroundCtx instead to ensure responses can be sent even when individual requests are canceled
• Improved error handling by checking and propagating errors from send operations rather than ignoring them, with failed sends triggering server shutdown via lspExit
• Enhanced context management by switching to WithCancelCause for better error tracking