Yes, this is the Live Templates feature of Delphi (one of my favorite productivity features, actually). A default set of templates gets installed in C:\Program Files (x86)\Embarcadero\Studio\21.0\ObjRepos\en\Code_Templates\Delphi as you discovered but remember that anything under C:\Program Files (x86) is read-only by default, so renaming or changing a Live Template from Delphi might not affect the actual file--or, as you encountered, you'd get the "Unable to create backup folder" error as Delphi tries to create a backup of the template before saving the new changes (which it will also fail to do unless you're running Delphi as Administrator).
In addition to the default templates mentioned above, any new ones you create are stored in C:\Users\<your username>\Documents\Embarcadero\Studio\code_templates\Delphi. Since your user folder IS writable, you can rename/modify/delete the templates there all you want.
You can turn off the auto-completion of the templates (which forces you to hit Ctrl+J to activate one) by going into Tools > Options > User Interface > Editor > Language, switching to the "Code Insight" tab, and unchecking "Auto complete" under "Code template completion".