The standard Save button will always close the page. If you do not want this, create a Action button with a Microflow that only commits the object.
With the property Close Button you can choose what button needs to be "Clicked" when the user uses the X on the top right of your pop-up. This is normally your cancel button. This ensures there is no difference whether the user clicks the X or the cancel button. This comes in handy when you have a custom MF Cancel button.
The Save on Close/Cancel property only has effect when your page is opened in content. Are you using a popup?