Time Machine is not the best tool for what you want to accomplish. If you are not able to use third-party software to do the job, then I suggest that you clone the virgin systems to separate volumes on your external drive. Start by erasing the external drive, then partition the drive into two volumes. If your external drive is an SSD, then use the APFS filesystem. If it's a normal HDD, then use Mac OS Extended, (Journaled).
Disk Utility for macOS Sierra- Partition a physical disk using Disk Utility
Partition Drives & Create APFS ‘Containers’ for Space Sharing with Disk Utility
Clone El Capitan and Later using Disk Utility
- Restart the computer. Immediately, at or before the chime press and hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
- Select Disk Utility from the Utility Menu, then click on the Continue button.
- Select the destination volume from the left side list.
- Select Restore from the Edit menu of Disk Utility. A panel will drop down.
- Select the source volume from the dropdown menu in the panel.
- Double-check that you got it right, then click on the Restore button.
You will repeat the above for each destination volume that you created on the external drive. Put one system on each volume. The clones may be used to restore a default setup on each student's machine.
A third-party software product exists to perform the same desired result. It is called Deep Freeze. You might wish to look into it because it was create from the ground up to accomplish exactly what you want. I suggest you look into it to find out if it is suited to your needs.
