Response:

It's just a matter of terminology: with System we refer normally to IT automation, with Business the domain. So basically we agree. The Business Analyst applies analytical skills to gather the requirements: "what" the business wants or needs. The solution can be paper based, people, process or guess what an IT system...
With Waterfall or Argile: it's just a matter of revisiting the Business requirements to ensure that they are still valid. Hope that helps.