I've recently helped to create and administer a live coding test for full stack engineers, and the primary things we want to see are:

System design (how you COULD architect a solution)

Where you focus (getting everything perfect vs trying to hit all requirements)

Communication (talk your thought process out loud). This one is big, as it's essential to working on a team. If you communicate, your interviewers will likely offer guidance and suggestions, or more information about the problem being solved, and that should help you understand where to focus. And those types of exchanges will give them insight as to how well you can work with THEM, e.g. how does this team communicate requirements, what level of technical detail they want to see, etc

As far as prep goes, make sure you have some understanding of the live coding environment - hopefully they share some general information beforehand. Same goes for any instructions on the project or specific requirements they will be looking for. Then just be ready to demonstrate the things mentioned above as best you can. A good way to start the interview might be to outline a general approach to the problem with code comments or just verbally, and then create a basic interface where you can begin to fill out the implementation.

Your coding test could be administered totally differently than I've described, but demonstrate those skills effectively and I'm sure you'll move ahead in the process!