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The way you normally do this for Word documents is Track Changes. Microsoft Word and Libre Office also have a built-in way to compare two documents, in case you didn't have Track Changes enabled. I assume if they're asking for document history, they want to see what changed when revising your draft, not in what order you pressed the keys. Although personally I tend to revise back the past part of the document as I'm writing instead of writing multiple drafts and this wouldn't really work for me. I guess another option is just to use set a timer and save a new version every so often. Maybe there's a trick you can do with scripting to make copies of your autosave file and then combine those into a format your teacher can understand.

Using Git is out of touch. It's incredibly unlikely that a language teacher understands Git or cares to learn how to use Git.

It sounds like your teacher probably has a way they expect students to do this, probably Google Docs or Office 365. Both of these have web versions so you could use them even if you're running Linux.