I think I figured this out. The Square item database is for your own internal categorization use. You have to set up the Weebly store by adding new categories (they can match your square store if you want, but don't have to) and then adding items to each new Weebly store category. This gives you flexibility to display items in your store differently than the way you may track them for your internal reports from the Square database categories.
To add items to a Category in Weebly store, in your Weebly store management page, go to CATEGORIES in left column, Add New or select an existing category, then click on + Choose Items to add items to that category. Make sure to hit the SAVE button at the upper right of the page when you're done, or it won't stick.
Regarding items that show up in the ITEMS category at the bottom of your store, rather than in one of your pre-selected Weebly store categories, I found that happens when I remove items from a Weebly category, but don't mark them as Unavailable or Hidden in the VISIBILITY section in Item > Edit Item. So in other words, it happens to items that were categorized, but are no longer categorized in the Weebly store.
