There are at least two other definitions I can think of: namely the smallest subset of the Dedekind reals which contains the rationals and is closed under limits of unmodulated Cauchy sequences (resp. under limits of modulated Cauchy sequences). I haven't thought about them too much, so maybe they collapse, and this doesn't really relate to your question, but your mention that the multi-valued Cauchy reals “are only the version that is Cauchy complete” made me think of this, as the reals I mention are, by definition, unmodulated-Cauchy-complete (resp. modulated-Cauchy-complete).
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