I would be glad to help. Based on the information you provided there are a few possibilities for this problem.
1. The file you shared may have been saved as a local copy by one of the people you shared with. They then shared that new file which resulted in a new share link.
2. Although a google account is not required for the share method you are using, if the person accessing your link is currently logged into a google account, this will be used when accessing the link and when that user makes an edit, it will record the username that was active when the edit was made.
So to resolve this issue for the teachers you are working with, I would create a new share link for the file you want to share, then instruct them to log out of any google account they may be using before accessing the file. You can instruct them to log out by telling them to visit google.com, then click the profile circle in the upper right corner and choose Sign Out from that menu.
It's possible that something else is the source of the problem, but without knowing exactly what steps the teachers are taking to access your file, there is no way to know for certain. Please let me know whether or not this resolves the issue for you.