Organizing and accessing contact information in Outlook.com
Just to double check, are you able to add a new contact or is the Save button also missing on that dialog?
To edit a Contact List open the People module, click on Your contact lists. To edit the contact list hover over it and click the Edit button (pencil looking icon). This is how it looks on my Outlook:
When you click Edit you get this dialog to add people to the list. Is the Save button missing on this dialog?