You're wrong here. What you describe is one possible use - in Torchbearer 2e that is called a Convince Crowd conflict.
But it can also be used to resolve an argument between two people, and in my experience that is the more typical use of it: in Torchbearer 2e these are called Convince conflicts and Negotiate conflicts.
(In case you're wondering why I mention TB2e, it's because it's a game by the same designers using many of the same basic principles and techniques, but with some refinement/development of the BW systems.)
I haven't quoted rules text in this post, but you can find it in the core rulebook, with further discussion in the Adventure Burner reprinted in the Codex, and if you review the free sample adventure
The Sword you'll also see there that Duel of Wits is (inter alia) a resolution system for an argument between two characters.