Another subtle detail is that while it's true that every real number corresponds to (and can be represented by) a Cauchy sequence of rationals, the very sequence itself might be undefinable.

The decimal numbers, for example, can be viewed as an infinite converging sum of powers of ten. Theoretically one could produce a description, but only a countable number of those could be written down in finite terms (some kind of finite recipe). So those finite ones could fall in a constructivist camp, but the ones requiring an infinite string to describe would, as far as I understand constructivism, not fall under being constructivist. To be clear, the finite string doesn't have other be explicit about how to produce the numbers, just that it is naming the thing and it can be derived from that. So square root of 2 names a real number and there is a process to compute out the decimals so that exists in a constructivist sense. But "most" real numbers could not be named.

If you are a constructivist, then you will supply direct proofs for your results as you reject indirect proof, proof by contradiction, law of excluded middle, and things of this nature.

The converse does not necessarily hold. Providing a direct construction of an object satisfying the field and completeness axioms (e.g. the Dedekind construction) does not necessarily mean that one is a constructivist. Indeed, one can use the Dedekind construction and still go on to prove many more results on top of it that still do rely on indirect proof and reductio ad absurdum.