Oh! I understand what happened now. In 'Customization' mode...the topmost item displayed instead of the presets is called "Rename Presets (Favorites)". When I was copying that to my toolbar, nothing would appear.
But right below that is another item called "Rename Presets (Non-Favorites)". When I copy that, then I'm able to see my 'Rename Groups' appear on the toolbar.
The only problem with this method is that when I have custom groups, it will add other individual (non-grouped) presets in a new group folder called 'Unspecified'.
If I don't use custom tab groups, that 'Unspecified' folder does not appear and the presets appear individually.
For this situation, the article you linked to is great. This specific section explains how to create new buttons by calling on those Rename presets:
To control how those buttons are grouped and displayed on a new toolbar, I think putting the buttons in a new menu is a good solution:
https://www.gpsoft.com.au/help/opus12/index.html#!Documents/Dropdown_Buttons.htm