Kobo expects the author to be in the form "FN LN". And if there are multiple authors for them to be separated by a comma. That means if you use "LN, FN", Kobo will treat this as two authors, "LN" and "FN". So, if you are sending the books to the device with the author in the style "LN, FN", you will get the result that you are seeing. And as this is how the devices are designed, it means that the device is working correctly.
Looking at the second screenshot, you have the author in the format "FN LN" and the sort author as "LN, FN". Or in this case "FN SN TN LN" and "LN, FN SN TN". (And I honestly could not have told you what "JRR" stood for. I don't think I have seen anything but his initials on books.) And without any other changes, calibre would sent "FN SN TN LN" as the author for the book and the device would use it. That suggests you have done something to sent the "author sort" field to the device rather than the author field.
The usual way to do this is to with a metadata plugboard. I think you have a metadata plugboard that is setting the author to the template "{author_sort}". That will set the author in the book to the author sort when it is sent to the device. If this is the case, you should remove it. That would solve the issue when you send new books to the device. But, I generally recommend using the following template in a plugboard for the author.
Code:
program: raw_list('authors', ', ')
This is so that books with multiple authors are written to the metadata in the book in the way that the Kobo devices want them.
That should solve the issue for all new books sent to the device. And if you have the metadata updating configured in whichever of the KoboTouch or KoboTouchExtended drivers you use, the authors for books on the device should be fixed the next time you connect the device.
As you mentioned the Kobo Utilities plugin, it doesn't really come into play here. It does have a method to update the metadata on the device. The metadata update has options to set the title in various ways. Just using the simple "Author" option should have fixed it. But, if you chose the option for using the author sort, or the plugboard, then it would not have solved the problem.