Jisho

Yes, verbs effectively have multiple て forms in the negative.

When you are using some kind of auxiliary with the て form, such as てください (please do x), or ておく (do x in advance), or てほしい (want someone to do x), or others, then you have to use ないで and not
なくて.

One other difference, is that if you want the negative verb to be used in kind of an adverbial sense, where it is describing the action in the next clause, then you have to use ないで. For instance, 走らないで教室に入る (enter the classroom without running).

This was just a pronounciation issue for Japanese people. The solution was the same as some kanji words like 火山 being "kazan" not "kasan" (plus it'd sound like kaa-san). Here 間違えないて is hard to pronounce so 間違えないで is easier. Japanese logic.

@JayDAshe yes, 間違えないて doesn't exist but 間違えなくて does. But its usage would be different from 間違えないで.