It seems ridiculous to me to define the real numbers by using the least-upper-bound property as an axiom, although this seems to be the most common way. The definition of the reals should formalise the way we approximate pi by a decimal expansion: 3, 3.1, 3.14 etc.
So the Cauchy sequence definition is much better, and I like the Dedekind cuts definition too. I don't at all understand how it makes sense to regard the least-upper-bound property of the reals as an axiom. The least-upper-bound property should be regarded as a theorem, not an axiom.
Why did the least-upper-bound-property-as-an-axiom approach become so prevalent? If you define real numbers that way, the correspondence between our intuitive sense of real numbers and the formalisation is so much less clear than with either Dedekind cuts or Cauchy sequences.