Learn English – “I’ll not” vs “I won’t” – when is which preferred

‘An A-Z of English Grammar and Usage’ by Leech and others suggests the following:

With be, use the contraction + not (e.g. That’s not right).

With have and modal auxiliaries, use the verb + n’t option, e.g. hasn’t, can’t.

The authors go on to say that forms such as She isn’t hungry, as opposed to She’s not hungry, are less common, and that forms such as I’ve not met him, as opposed to I haven’t met him, are much less common.

The authors don’t say on what basis they make their recommendations, but they are likely to be based on frequency of use. There doesn’t seem to be any difference of meaning or emphasis. I would just add myself that a contraction such as I’ve not met him sounds more formal, perhaps because of its lower frequency.

Both of your examples #3 and #4 mean more or less the same thing. #3 is just a different and less common style that sometimes works better in a longer sentence, possibly when comparing with other negations. For example:

She is not often sad (so it's unusual to see her sad today)

She is often not sad (but not happy either).

The reason is that it shifts the negation from the adverb often to the adjective sad, not-often vs. not-sad.

In a similar way your example shifts the negation from the adjective to the noun. The first says the mistake is of type not-X, while the second says it's not a type-X-mistake. As I said, it's a difference that, in this case, makes no difference in meaning, but might in a different context:

This is a not-uncommon procedure (so we should have nothing to worry about)

This is not an uncommon procedure (but it's not a particularly common procedure either).