So in order to stop using heading styles, let's say I have a book with chapters. I only want the chapter numbers and titles to show up in the TOC.
I use Header 2 from the home screen.

I think this is the point I must be going off track. When I begin a chapter, I just start writing and the triangles show up.

I need to know, if you don't mind, exactly what to do at this point so the header tri's don't show up.

Select the text you do not want to appear in the TOC, do not want to be collapsible, and apply the Normal or Body Text style to it or some other style that is not on an outline level other than body text.