What you describe as a "user story" (the "as a , I want <goal/desire> so that " part) is really the card, which is one-third of a user story. The other two parts are conversation and confirmation. Together, these are known as the Three Cs.
The card is a starting point for a conversation, but it isn't the end. The card allows you to order the work in some kind of backlog, remember the work for the future, and remind you who needs to be involved in the conversation. By having a conversation, you would learn about other things that stakeholders need.
There are different ways to capture the results of the conversation. If you're using an electronic tool for capturing user stories, you may be able to attach things like conversation or meeting notes, wireframes and mockups, decision tables, data dictionaries, roles and permissions matrices, and more. If you have physical cards for user stories, you may store this information in some kind of wiki or CMS and have a way to find the information relevant to the user story.
Exactly how and what you record about the conversation depends on the needs of the team and the context of the organization. This is the Agile Software Development values of "individuals and interactions over processes and tools" and "working software over comprehensive documentation".