Adding New Component to Cloud Library Saves it as a Graphic, not a Component

Setup/Problem: I had, long ago, created an enormous working file with 50+ artboards and from that created a single "master" library of colors, text styles, and components. Let's call this Library A. I wanted to clone Library A and "start fresh" - using that original, very sloppy library as a foundation for building out a fresh one to use across multiple files - one that already contained a ton of the assets I was and would be working with more.

Problem: Working with an existing cloud library full of elements seems incredibly simple, but the process of updating/adding to it seems anything but. Document assets vs. Existing Library is specifically incredibly confusing and I cannot find a good explanation for how to reconcile them. 

Process/Confusion: 

1. I started a new document entirely
2. I duplicated Library A and started renaming elements, deleting duplicates, etc. Just cleanup to have to a nice solid start. Lets call this LIBRARY B
3. I wanted to add new components to to Library B, so I made some in this new document, which was "using" Library B, and converted them to components. This immediately, and very confusingly added the new components to the "document assets library" - switching the library pane on the left to show me this.
4. I looked up content on this message board to see what I needed to do, and the only solution I found involved making something in the document, and with Library B open, using the + button on the bottom to "add element." (really? no option in the context/right click menu?)
5. Anything that I had converted into a component initially, when added as an element into the library, is saved simply as a "graphic" - not a component that I can drag onto the document and modify/have a master/etc.