Revert "refactor(lsp): simplify client tracking"

This reverts commit 3f238b39cf.

runtime/lua/vim/lsp/client.lua

@@ -761,7 +761,7 @@ function Client:_request_sync(method, params, timeout_ms, bufnr)
   return request_result
 end
 
---- @package
+--- @private
 --- Sends a notification to an LSP server.
 ---
 --- @param method string LSP method name.