A step that is missing is picking your strongest argument. A mistake that I often see made is that people have 50 reasons why that person is wrong but usually 49 of them are fairly weak and could be debated. You are generally better off picking your one killer argument and just bolstering that. Otherwise you get into a position where they seem like they are winning because they knocked aside a bunch of your less meaty arguments.
Julie Zhuo with a framework you can follow when disagreeing with others. I think it's helpful to understand which "hat" you want to put on when providing feedback: are you the (1) Owner of the task (the one who should get it done and care for it in the long run, the (2) Consultant (an expert with no direct ownership) or the (3) Preacher (you have some interest in the domain, but you neither the owner nor the expert). Understanding your role is where I'd start, as it will help to set the context in which you see the contradictions with the other person.