logic - 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?

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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.

Answer

Your question is essentially a philosophical one. When ordinary people talk about number, they have a very definite concept in mind. Mathematicians are ordinary people, but have a professional obligation to give logical justifications for their use of the number concept. So when they are concerned about logical foundations, mathematicians give existence proofs for the various number systems by constructing explicit witnesses for systems satisfying the required properties. When they go back to their day-to-day mathematical work, mathematicians forget about the details of these existence proofs and happily work as if $\Bbb{R}$ were a subset of $\Bbb{C}$ and $\sqrt{2}$ were an object that belongs to $\Bbb{R}$, satisfies $x^2 = 2$, but isn't itself a set or a sequence.

There is a seminal paper by Benacerraf What Numbers Could Not Be that anyone interested in the philosophical foundations of mathematics should read.