So when you create a nintex form on a list, you are creating the form for Display and Edit and New. You can modify how items appear on form by editing the control settings, and form settings. For controls, under Advanced, look for Control Mode. In the Visible section, you can use the run time function of isEditMode, or isDisplayMode, or isNewMode.
I'd like to have the Nintex Form in Display mode different from the New
and Edit modes. For instance, to have the controls in different place or
to customize the Close Button.
That makes sense - but currently this type of functionality isn't supported out of the box in Nintex Forms. The new, edit, and display all use the same form layout. Really the only thing you could affect is what is displayed or disabled in a given mode, but not the placement of the controls on the form. The only way I know of would be to use some kind of rules based trickery, where all the panels with duplicated information existed but were hidden/shown at the various display modes but this of course has its own pitfalls (maintainability..etc..)
Here is how to use hide rules in case you want to look into this type of visual trickery:
If you are talking about a button that you put on the form, then you can double click on your button and open the configuration screen, you can type whatever text you want to appear in the Button Label field.