To save the selected content without saving changes to the InDesign document or any of its linked files, select the content in the layout and choose Edit > InCopy > Save Content.

To save every content file checked out to you, choose File > Save. This type of save does not automatically update assignment files or checked-out content files in InCopy. However, the checked-out files in InCopy will show the Out Of Date icon  in the Assignments panel.

To save the InDesign document under a new name with links to any existing InCopy files, choose File > Save As. This type of save creates a situation in which one or more InCopy files are linked to two InDesign documents.

To save a copy of the currently active document, choose File > Save A Copy. You can choose to give the copy a new name and location with links to any existing InCopy files.