I think the truth is that nobody cares. I mean, you care about such matters a little bit while learning how set theory can be used as a foundation for mathematics, but it soon ceases to be of any importance. In practice, the one important thing about n-tuples is the relation between the n-tuple and its components, i.e., the fact that two n-tuples are the same if and only if they have the same components in the same order.
If you don't learn to stop worrying about such minutiae, you will have plenty more troubles as you learn about number systems. What is the number 3, really? It could be the ordinal {0,1,2} (i.e., {∅,{∅},{∅,{∅}}}), or it could be the integer 3 represented as an equivalence class {(m,n):m=n+3} of ordered pairs of ordinals, or it could be the rational number 3 represented as an equivalence class {(p,q):p=3q, q≠0} of ordered pairs of integers, or it could be the real number 3 represented by whatever your method of defining the real numbers happen to be, or it could even be the complex number represented as a pair of real numbers (3,0) … I hope you get my drift. Every time you expand the number system, and often when you generalize some notion or other, the new contains an isomorphic copy of the old and nobody cares to distinguish between copies.
This practice of identification has its dangers, of course, so it's good that you worry about such things a bit while learning, but expect such matters to recede into the background in order to make room for more important things.
(For what it's worth, I think the method in your second paragraph is good, but having two kinds of ordered pairs should soon stop bothering you.)
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