Oh yeah absolutely and formality is generally more important in German than it is in English (e.g. there is a distinction between formal and informal 'you').
However, to me it's still weird in this context for the teacher to tell her it is incorrect, the teacher should have just told her it's less formal rather than saying it is incorrect altogether (which it isn't). The sentence isn't 'informal' in the sense that it is incorrect slang or anything, it's just the shorter more casual way of saying it (but honestly there isn't even much of a difference between the two in terms of formality in my opinion?) Like I said, it's pretty much like "I'm Jade" instead of "My name is Jade" in English.
This isn't a Lingoda thread, so it's whatever and maybe Jade just misunderstood but I just thought it was a strange moment because the way it came across was that the teacher incorrectly 'corrected' her