Which saying is correct?

Writing in Google I got these results:

• "It's no skin off my teeth" -> About 36,300 results
• "It's no skin off my nose" -> About 449,000 results

But on the dictionary I found both and they seem to have different meanings, even if the first one uses by the skin of and not no skin off:

by the skin of one's teeth by a very narrow margin; barely : I only got away by the skin of my teeth. [ORIGIN: from a misquotation of Job 19:20: “I am escaped with the skin of my teeth” (i.e., and nothing else). Current use reflects a different sense.]

it's no skin off my nose (or off my back) informal (usually spoken with emphasis on “my”) used to indicate that one is not offended or adversely affected by something: it's no skin off my nose if you don't want dessert.

You are correct. By the skin of your teeth, means very close. It missed me by the skin of my teeth. Your relative seems to be confusing metaphors. Some similar confused metaphors that someone in my family uses: Don't kick a gift horse in the mouth and it's six of one seven of the other.