I would also assume that your school has rules about this.
Continuing: whenever I go visit a hospital and have a medical student on rounds or shadowing, they just have their short white coat on without any other identifying information on it. No nametag, no sign saying "medical student". They do wear their badge/ID as everyone else should. Most of the time the white coats will have a patch with the school's seal on it so I can tell which school they study at (because it's not just one school). In contrast, the dental students have their names and school embroidered on their scrubs which they have to wear all the time; in these cases, there is no embellishment with titles or degrees. I got my own white coat after I graduated with my doctorate, and it has my full name, my degree, and the department I worked for.
I would be extremely surprised if -- as a third-year student -- you have full leeway on what to do. Every clinic will have its rules, but usually the school has a say in terms of your dress code and specifics on what should be on your white coat during your clerkships.