If it’s not possible to do a CSS hack or another admin side edit to show the Name instead of the Full Name, perhaps there could be a future admin setting at https://yourforum.com/admin/site_settings/category/groups to toggle between displaying the Name or Full Name value. It might look like this:
I thought about that myself before creating the topic. It would work OK for the group listing page, but not so well for the group detail page.
Because we have so many groups and plan to add even more, our community appreciates a short description of a group so they don’t have to read the full description when they’re considering joining.
For example, if I made our Arts & Crafts group keep the same Name value and change the Full Name value to “ArtsAndCrafts” as well, you’d miss the current “Artist and Crafters of All Mediums, Forms, and Disciplines” short description.