The image you posted shows the steps for updating FROM the styles library, which is the opposite of saving to the styles library. You have to select the option to save to the styles library (shown below):
If this command is not available, then your project file probably has the styles library set to read-only status. You will need to change this:
Please click "Accept as Solution" if this response answers your question.
Make sure that the project file is not read only (right click on explorer), that is not vaulted (checked in) and that you only have it open on one Inventor session. Others need to close that project if it's on a shared network drive.