I'm glad you've found some help, but please, find another teacher. This one is not the right one for you. He may be a brilliant musician, or even a brilliant teacher, but he's not teaching you. His teaching style (dogmatic, teacher is always right) and your learning style (ask questions, need more than rote "do this") are completely out of synch.
Learning music should be a joy not an exercise in misery.
I'll try to answer the question you asked, but didn't get an answer to. Music, especially traditional music, is highly variable. I recommend listening to the same tune played by different musicians -- even by the same musician on different occasions. You'll still recognize the tune, but it won't be note-for-note identical to any other performance.
Key -- a melody is relative to its key. We can change the key without changing the melody. Whether you hear Wabash Cannonball in G or D, you'll still easily recognize the tune. We change keys for a lot of reasons. Sometimes it's forced by the instrument. If all I have is my D penny whistle, then G, D and C (and the relative minors) are about all I can handle. We change keys for singers a lot. We can change from major to minor (or the other way) to change the mood of a piece.
We can change the notes and rhythm as well while still making the tune recognizable. It's amazing how flexible the human ear (mind, really) can be. We're programmed to find patterns and patterns we will find.
There's no firm line that says "you've changed 22.7% of the notes, so it's now a different tune." One alteration that you find in just about every traditional musical performance is adding or subtracting ornamentation. Grace notes, turns, trills are all alterations that leave the tune itself almost untouched. Listen to a traditional fiddler play a tune and hear the variations from one repetition to another.
Although classical music tends to be more rigid in performance, in part because it works from written music rather than by ear, you still get alterations to a melody that still leave it recognizable. Try Mozart's "Twelve Variations on Vous dirai-je, Maman" K. 265/300e.
He takes a very familiar tune and puts it through many variations. This includes adding notes and ornamentation as well as changing the key from major to minor. You'll still recognize the tune.
A more humorous example would be Tom Lehrer's version of Clementine.