I think that occurred post WW2 when the commissioning requirement made the cadets Naval officers in training similar to today. Midshipmen on campus, cadet at sea.
There are different legal meanings.
1. USNR E-1 appointed to Mishipman in the Inactive Ready Reserve as part of the Strategic Sealift Midshipman Program for the purposes of the Navy.
2. A school term for what they call their civilian students for the purposes of the federally operated [DoT/MarAd] maritime academy. [ have yet to see an actual US Code law or CFR that specifically says a "regiment" is required.
Students sign 2 types of documents [navy and marad] that are coordinated, but separate.
...and then there is the term used for the Coast Guard for the role on a ship.
It gets very grey and confusing.
The tough part is what exactly is the status of the students while attending the school....I have never gotten a concrete legally acceptable reason.
I have heard Cadet, Cadet-Midshipman, and Midshipman. Away from school, and during Sea Year, I don't know that I ever heard anything but "Cadet". I am trying to remember what my original Z-Card stated. Midshipman is a term used for a naval officer trainee. Cadet is common for military academies, except Navy, where they are all Midshipmen. Cadet is also a common term world wide for merchant ship officer apprentices.