What "real numbers" (elements in $\mathbb R$), are people referring to? To define mathematical objects, it seems one defines them in terms of other mathematical objects.
However various mathematical objects have different "definitions".
E.g. it seems people "construct" the real numbers (they use objects other than the real numbers, to create an algebraic structure isomorphic to what others consider the real numbers) and then define those as the real numbers. Yet this appears ambiguous to me.
If one were to refer to the "real numbers" would they then be referring to Dedekind Cuts? 
Or would they be referring to equivalence classes of Cauchy Sequences?
Or to some other "construction" entirely? 

Here are my best two guesses as to what people mean:
$1.$ They are simply referring to any structure isomorphic to the (unique) complete ordered field.
$2.$ They are referring to an equivalence-class$^{*}$ of all structures that are isomorphic to the (unique) complete ordered field.
$*\small(\text{Not an equivalence class in the proper sense with sets as an such object cant be a set by Russell's paradox)}$
The reason $1$ would not result in ambiguity is because we often never need to make use of their original "definitions" per se. Because we can embed the rationals, integers etc. inside the real numbers. However $2$ would avoid this problem entirely as we have only a single definition.

I'd go for (1), slightly modified. If you've just constructed the real numbers in some book or paper, using, say, Dedekind cuts, then in that context a real number is a Dedekind cut, and you use properties of Dedekind cuts to prove theorems about real numbers.
Since there is a unique ordered field (up to unique isomorphism) and your construction provides one, your theorems will be true for anyone else's "real numbers".
Edit: this is essentially @AsafKaraglia 's answer to the duplicate, but much less informative.
