Is it proper to say (as I keep reading) that the real numbers are "equal" to the equivalence classes of Cauchy sequences in the completion of the rational numbers. Yes, there is a one-to-one correspondence with the limits of the sequences, but "equal to" seems too strong. Am I correct in this?
There are several possible construction of the real numbers that, in the usual framework of set theory (ZFC or equivalent) give rise to isomorphic structures.
It doesn't matter how the real numbers are constructed, so long as they are an archimedean ordered field $F$ which is Cantor-complete (if $S$ and $T$ are non empty subsets of $F$ such that $s\le t$, for all $s\in S$ and $t\in T$, then there exists $r\in F$ such that $s\le r\le t$, for all $s\in S$ and $t\in T$).
Two such structures are isomorphic in a strong sense: there is a unique field isomorphism between them. In particular they can't be told apart by just using properties of ordered fields. Moreover there is a unique embedding of the rational numbers in such a field.
The fact that one is the set of equivalence classes of Cauchy sequences in the rationals and the other is, maybe, the set of Dedekind cuts of the rationals is completely irrelevant, as far as the theory of real numbers is concerned. One just uses their properties summarized above.
If you like to think that a real number is an equivalence class of Cauchy sequences, you're welcome, but it really adds nothing to your understanding of the real numbers. The constructions are important because with them we know we're talking about something.
Such a field can be given a uniformity which makes it into a topological field (the operations are continuous), so a notion of Cauchy net can be given (that doesn't appeal to the real numbers) and it can be proved that this uniform space is complete, that is, every Cauchy net converges. In particular (using the unique embedding of the rationals mentioned above), every Cauchy sequence of rationals converges.