> several of the definitions in the link feel constructivist, i.e. they describe constructions of of real numbers.

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.

Interestingly, constructive mathematics cannot prove that the Cauchy and Dedekind constructions are isomorphic:

"As often happens in an intuitionistic setting, classically equivalent notions fork. Dedekind reals give rise to several demonstrably different collections of reals when only intuitionistic logic is assumed"

https://arxiv.org/pdf/1510.00641