It depends on the GUI.

For photo editing, the interactions with the photo directly require fine control in two dimensions with the cursor.

Many GUIs are just interfaces to modal switches, or a categorical variable with usually a finite number of options.

Sort of (depends on terminology): A form is a set of controls laid out in 2D. But they are ordered and discrete so you can just ignore their positioning in x,y and instead navigate in their order instead (e.g tabbing through controls) making it a 1D input!

By “2D input” in this context I mean input where there isn’t also such a navigation option and you must input one or more coordinates and there is no way to enter the coordinates as numbers. For example the input surface (image) in a drawing program like Microsoft paint.

yes. But text UIs are, arguably, infinite-dimensional (because language is). Thus a 2D input problem is a severely limited particular case, that allows for very specific and useful optimizations helped by a mouse, pad or trackball.