Hi, sorry for replying so late. Are you using an old GB Studio 2 version? The README file explains that older GB Studio 2 versions don't update the UI part of VRAM. Either update to a newer version or load the letters_vram.png sprite after the text input screen. You should also modify it to use your own border and cursor.
Viewing post in SodoDev's Text Input System for GB Studio 2 comments
No, check the zip. There should be a sprite called letters_vram.png. Make a blank scene with an actor that uses that sprite after the name input screen. It can be hidden.
That file should have a border and a cursor at the end. Modify that part and replace it with your border and cursor.
Also, there are several GB Studio 2 versions. You might have beta1, which has this problem.